15 January 2014, Volume 30 Issue 1

    

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  Analysis on a Superlinearly Convergent Augmented Lagrangian Method

  Ya Xiang YUAN

  Acta Mathematica Sinica. 2014, 30(1): 1-10. https://doi.org/10.1007/s10114-013-2740-9

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  The augmented Lagrangian method is a classical method for solving constrained optimization. Recently, the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low rank matrix optimization problems. However, most Lagrangian methods use first order information to update the Lagrange multipliers, which lead to only linear convergence. In this paper, we study an update technique based on second order information and prove that superlinear convergence can be obtained. Theoretical properties of the update formula are given and some implementation issues regarding the new update are also discussed.
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  Automorphism Groups of Pseudo-real Riemann Surfaces of Low Genus

  Emilio BUJALANCE, Antonio F. COSTA

  Acta Mathematica Sinica. 2014, 30(1): 11-22. https://doi.org/10.1007/s10114-013-2420-9

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  A pseudo-real Riemann surface admits anticonformal automorphisms but no anticonformal involution. We obtain the classification of actions and groups of automorphisms of pseudo-real Riemann surfaces of genera 2, 3 and 4. For instance the automorphism group of a pseudo-real Riemann surface of genus 4 is either C₄ or C₈ or the Fröbenius group of order 20, and in the case of C₄ there are exactly two possible topological actions. Let M_PR,g^K be the set of surfaces in the moduli space M_g^K corresponding to pseudo-real Riemann surfaces. We obtain the equisymmetric stratification of M_PR,g^K for genera g = 2, 3, 4, and as a consequence we have that M_PR,g^K is connected for g = 2, 3 but M_PR,4^K has three connected components.
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  Fluctuation Limits of the Single Point Catalytic Super-stable Processes

  Li WANG

  Acta Mathematica Sinica. 2014, 30(1): 23-34. https://doi.org/10.1007/s10114-013-2465-9

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  We prove a fluctuating limit theorem of a sequence of super-stable processes over R with a single point catalyst. The weak convergence of the processes on the space of Schwartz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional stable process.
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  Operator Jensen's Inequality on C^*-algebras

  Xin LI, Wei WU

  Acta Mathematica Sinica. 2014, 30(1): 35-50. https://doi.org/10.1007/s10114-013-2065-8

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  A real continuous function which is defined on an interval is said to be A-convex if it is convex on the set of self-adjoint elements, with spectra in the interval, in all matrix algebras of the unital C^*-algebra A. We give a general formation of Jensen's inequality for A-convex functions.
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  Entropy Numbers of Besov Classes of Generalized Smoothness on the Sphere

  He Ping WANG, Kai WANG, Jing WANG

  Acta Mathematica Sinica. 2014, 30(1): 51-60. https://doi.org/10.1007/s10114-013-3044-9

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  We investigate the asymptotic behavior of the entropy numbers of Besov classes BB_p,θ^Ω (S^d-1) of generalized smoothness on the sphere in L_q(S^d-1) for 1 ≤ p, q, θ ≤ ∞, and get their asymptotic orders. We also obtain the exact orders of entropy numbers of Sobolev classes BW_pr (S^d-1) in L_q(S^d-1) when p and/or q is equal to 1 or ∞. This provides the last piece as far as exact orders of entropy numbers of BW_pr (S^d-1) in L_q(S^d-1) are concerned.
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  An Integral Representation for the Weighted Geometric Mean and Its Applications

  Feng QI, Xiao Jing ZHANG, Wen Hui LI

  Acta Mathematica Sinica. 2014, 30(1): 61-68. https://doi.org/10.1007/s10114-013-2547-8

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  By virtue of Cauchy's integral formula in the theory of complex functions, the authors establish an integral representation for the weighted geometric mean, apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function, and find a new proof of the well-known weighted arithmetic-geometric mean inequality.
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  A Central Limit Theorem for Branching Brownian Motion with Random Immigration

  Hong Yan SUN

  Acta Mathematica Sinica. 2014, 30(1): 69-78. https://doi.org/10.1007/s10114-013-2148-6

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  We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law, where the immigration is determined by another branching Brownian motion. The limit is a Gaussian random measure and the normalization is t3/4 for d = 3 and t1/2 for d ≥ 4, where in the critical dimension d = 4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.
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  Gromov Hyperbolicity of Periodic Planar Graphs

  Alicia CANTÓN, Ana GRANADOS, Domingo PESTANA, José Manuel RODRÍGUEZ

  Acta Mathematica Sinica. 2014, 30(1): 79-90. https://doi.org/10.1007/s10114-013-2370-2

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  The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.
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  A Note on List Edge and List Total Coloring of Planar Graphs without Adjacent Short Cycles

  Hui Juan WANG, Jian Liang WU

  Acta Mathematica Sinica. 2014, 30(1): 91-96. https://doi.org/10.1007/s10114-013-2419-2

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  Let G be a planar graph with maximum degree Δ. In this paper, we prove that if any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4} in G, then the list edge chromatic number χ_l'(G) = Δ and the list total chromatic number χ_l" (G) = Δ+1.
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  Index Computations in FJRW Theory

  Ai Jin LIN

  Acta Mathematica Sinica. 2014, 30(1): 97-118. https://doi.org/10.1007/s10114-013-2649-3

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  We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev spaces.
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  Complete Convergence and Complete Moment Convergence for Martingale Difference Sequence

  Xue Jun WANG, Shu He HU

  Acta Mathematica Sinica. 2014, 30(1): 119-132. https://doi.org/10.1007/s10114-013-2243-8

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  In the paper, we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As an application, a strong law of large numbers for martingale difference sequence is obtained.
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  On Solution Sequences of the Singular Liouville Equations with an Integral Constraint

  Yong LUO

  Acta Mathematica Sinica. 2014, 30(1): 133-150. https://doi.org/10.1007/s10114-013-1583-8

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  In this paper, we analyze the blow-up behavior of sequences {uk} satisfying the following conditions
  ?Δuk = |x|^2α_kV_ke^uk in Ω, (0.1)
  where Ω ⊂⊂ R², V_k → V in C¹, |∇V_k| ≤ A, 0 < a ≤ V_k ≤ b, 0 ≤ α_k → α∈ (0,∞), and
  ∫_Ω |x|^2α_ke^ukdx ≤ Λ₁. (0.2)
  Furthermore, we assume that there exists some q ∈ (1, 2) such that
  r^q?2∫_Br(p) |∇uk|^{qdx ≤ Λ₂ (0.3)
  for any B_r(p)} ⊆ Ω^. As a result, we give a new proof of the concentration-compactness theorem for the mean field equation^.
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  Bounded and Compact Toeplitz Operators on the Weighted Bergman Space over the Bidisk

  Si Bo DIAO, Tao YU

  Acta Mathematica Sinica. 2014, 30(1): 151-162. https://doi.org/10.1007/s10114-013-3067-2

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  For α >-1, let dvα denote the weighted Lebesgue measure on the bidisk and μ a complex measure satisfying some Carleson-type conditions. In this paper, we show a sufficient and necessary condition for the Toeplitz operator T_μ^α to be bounded or compact on weighted Bergman space L_a¹(dvα).
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  Total Duration of Negative Surplus for a Brownian Motion Risk Model with Interest

  Wei WANG, Jing Min HE

  Acta Mathematica Sinica. 2014, 30(1): 163-168. https://doi.org/10.1007/s10114-013-2008-4

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  In this paper, we consider the Brownian motion risk model with interest. The Laplace transform of the first exit time from the upper barrier before hitting the lower barrier is obtained. Using the obtained result and exploiting the limitation idea, we derive the Laplace transform of total duration of negative surplus.
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  Laguerre Isoparametric Hypersurfaces in Rⁿ with Two Distinct Non-zero Principal Curvatures

  Yu Ping SONG

  Acta Mathematica Sinica. 2014, 30(1): 169-180. https://doi.org/10.1007/s10114-013-1517-5

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  An umbilical free oriented hypersurface x : M → Rⁿ with non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C =∑_iC_iω_i = ∑_iρ^-1(E_i(log ρ)(r -r_i)-E_i(r))ω_i vanishes and Laguerre shape operator S = ρ^-1(S^-1-rid) has constant eigenvalues. Here ρ =∑_i(r -r_i)², r = (r1+r2+…+r_n-1)/n-1 is the mean curvature radius and S is the shape operator of x. {E_i} is a local basis for Laguerre metric g = ρ²Ⅲ with dual basis {ω_i} and Ⅲ is the third fundamental form of x. In this paper, we classify all Laguerre isoparametric hypersurfaces in Rⁿ(n > 3) with two distinct non-zero principal curvatures up to Laguerre transformations.
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  Binding Numbers for Fractional ID-k-factor-critical Graphs

  Si Zhong ZHOU

  Acta Mathematica Sinica. 2014, 30(1): 181-186. https://doi.org/10.1007/s10114-013-1396-9

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  Let G be a graph, and k ≥ 2 be a positive integer. A graph G is fractional independentset-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G-I has a fractional k-factor for every independent set I of G. The binding number bind(G) of a graph G is defined as
  bind(G) = min {|NG(X)|/|X|: ø≠X ⊆ V (G),NG(X) = V (G)}.
  In this paper, it is proved that a graph G is fractional ID-k-factor-critical if n ≥ 6k-9 and bind(G) >((3k-1)(n-1))/(kn-2k+2).