15 April 2014, Volume 30 Issue 4

    

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  The Defocusing Energy-supercritical Hartree Equation

  Ji Qiang ZHENG

  Acta Mathematica Sinica. 2014, 30(4): 547-566. https://doi.org/10.1007/s10114-014-3299-9

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  In this paper, we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu-(|x|^-γ* |u|²)u = 0 with γ > 4 in dimension d > γ. We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, u ∈ L_t^∞ (I; _x^s_c (R^d)) with x_c^s:= (γ/2)-1 > 1, then u is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation (NLW) and nonlinear Schrödinger equation (NLS). We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.
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  Yetter-Drinfeld Modules over Weak Braided Hopf Monoids and Center Categories

  José Nicanor ALONSO ÁLVAREZ, Ramón GONZÁLEZ RODRÍGUEZ, Carlos SONEIRA CALVO

  Acta Mathematica Sinica. 2014, 30(4): 567-590. https://doi.org/10.1007/s10114-014-3240-2

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  In this paper, we introduce several centralizer constructions in a monoidal context and establish a monoidal equivalence with the category of Yetter-Drinfeld modules over a weak braided Hopf monoid. We apply the general result to the calculus of the center in module categories. Keywords Weak (braided) Hopf monoid, Yetter-Drinfeld module, center
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  On a Relation between the Universal Teichmüller Space and the Grunsky Operator

  Masahiro YANAGISHITA

  Acta Mathematica Sinica. 2014, 30(4): 591-600. https://doi.org/10.1007/s10114-014-2761-z

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  Being a Strebel point gives a sufficient condition for that the extremal Beltrami coefficient is uniquely determined in a Teichmüller equivalence class. We consider how Strebel points are characterized. In this paper, we will give a new characterization of Strebel points in a certain subset of the universal Teichmüller space by a property of the Grunsky operator.
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  f-Class Two Graphs Whose f-Cores Have Maximum Degree Two

  Xia ZHANG, Gui Ying YAN, Jian Sheng CAI

  Acta Mathematica Sinica. 2014, 30(4): 601-608. https://doi.org/10.1007/s10114-014-3145-0

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  An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v ∈ V (G) at most f(v) times. The f-core of G is the subgraph of G induced by the vertices v of degree d(v) = f(v)max_v∈V (G){ d(v)/f(v)}. In this paper, we find some necessary conditions for a simple graph, whose f-core has maximum degree two, to be of class 2 for f-colorings.
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  Asymptotic Estimates on the Time Derivative of Φ-entropy on Riemannian Manifolds

  Bin QIAN

  Acta Mathematica Sinica. 2014, 30(4): 609-618. https://doi.org/10.1007/s10114-014-2720-8

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  In this note, we obtain an asymptotic estimate for the time derivative of the Φ-entropy in terms of the lower bound of the Bakry-Emery Γ₂ curvature. In the cases of hyperbolic space and the Heisenberg group (more generally, the nilpotent Lie group of rank two), we show that the time derivative of the Φ-entropy is non-increasing and concave in time t, also we get a sharp asymptotic bound for the time derivative of the entropy in these cases.
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  Wavelet Transform and Radon Transform on the Quaternion Heisenberg Group

  Jian Xun HE, He Ping LIU

  Acta Mathematica Sinica. 2014, 30(4): 619-636. https://doi.org/10.1007/s10114-014-2361-y

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  Let Q be the quaternion Heisenberg group, and let P be the affine automorphism group of Q. We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L²(Q). A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on Q. A Semyanistyi-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth. In addition, we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.
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  Calderŗn-Zygmund Operators in the Bessel Setting for All Possible Type Indices

  Alejandro J. CASTRO, Tomasz Z. SZAREK

  Acta Mathematica Sinica. 2014, 30(4): 637-648. https://doi.org/10.1007/s10114-014-2326-1

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  We show that many harmonic analysis operators in the Bessel setting, including maximal operators, Littlewood-Paley-Stein type square functions, multipliers of Laplace or Laplace-Stieltjes transform type and Riesz transforms are, or can be viewed as, Calderŗn-Zygmund operators for all possible values of type parameter λ in this context. This extends results existing in the literature, but being justified only for a restricted range of λ.
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  Nowhere-zero 15-Flow in 3-Edge-connected Bidirected Graphs

  Er Ling WEI, Wen Liang TANG, Dong YE

  Acta Mathematica Sinica. 2014, 30(4): 649-660. https://doi.org/10.1007/s10114-014-2028-8

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  It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero k flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet's conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.
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  Vertex-disjoint K₁ + (K₁ ∪K₂) in K_1,4-free Graphs with Minimum Degree at Least Four

  Yun Shu GAO, Qing Song ZOU

  Acta Mathematica Sinica. 2014, 30(4): 661-674. https://doi.org/10.1007/s10114-014-2027-9

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  A graph is said to be K_1,4-free if it does not contain an induced subgraph isomorphic to K_1,4. Let k be an integer with k ≥ 2. We prove that if G is a K_1,4-free graph of order at least 11k-10 with minimum degree at least four, then G contains k vertex-disjoint copies of K₁ + (K₁ ∪ K₂). Keywords Forbidden graphs, Vertex-disjoint subgraphs, Minimum degree
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  Γ-inverses of Bounded Linear Operators

  Xiao Ming XU, Hong Ke DU, Xiao Chun FANG

  Acta Mathematica Sinica. 2014, 30(4): 675-680. https://doi.org/10.1007/s10114-013-2552-y

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  Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H. For A, P and Q in B(H), if there exists an operator X ∈ B(H) such that APXQA = A,XQAPX = X, (QAPX)^* = QAPX and (XQAP)^* = XQAP, then X is said to be the Γ-inverse of A associated with P and Q, and denoted by A_P,Q⁺ . In this note, we present some necessary and sufficient conditions for which A_P,Q⁺ exists, and give an explicit representation of A_P,Q⁺ (if A_P,Q⁺ exists).
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  Hyperbolic Metric and Normal Family

  Cai Yun FANG, Xue Cheng PANG

  Acta Mathematica Sinica. 2014, 30(4): 681-688. https://doi.org/10.1007/s10114-013-2344-4

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  In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.
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  Characterization of Arithmetic Functions that Preserve the Sum-of-squares Operation

  Bojan BAŠIĆ

  Acta Mathematica Sinica. 2014, 30(4): 689-695. https://doi.org/10.1007/s10114-014-3112-9

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  We characterize all functions f : N → C such that f(m² + n²) = f(m)² + f(n)² for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f(n) = ±n (subject to some restrictions on when the choice of the sign is possible) and f(n) = ±(1/2) (again subject to some restrictions on when the choice of the sign is possible). Keywords Arithmetic function, functional equation, sum of squares
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  On All Fractional (a, b, k)-Critical Graphs

  Si Zhong ZHOU, Zhi Ren SUN

  Acta Mathematica Sinica. 2014, 30(4): 696-702. https://doi.org/10.1007/s10114-014-2629-2

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  Let a, b, k, r be nonnegative integers with 1 ≤ a ≤ b and r ≥ 2. Let G be a graph of order n with n > ((a+b)(r(a+b)-2))+ak/a . In this paper, we first show a characterization for all fractional(a, b, k)-critical graphs. Then using the result, we prove that G is all fractional (a, b, k)-critical if δ(G) ≥ ((G-1)b²)/a+k and |N_G(x₁)∪N_G(x₂)∪…N_G(x_r)| ≥ ((bn+ak)/(a+b)) for any independent subset {x₁, x₂, ...,x_r} in G. Furthermore, it is shown that the lower bound on the condition |N_G(x₁)∪N_G(x₂)∪…∪N_G(x_r)| ≥ ((bn+ak)/(a+b)) is best possible in some sense, and it is an extension of Lu's previous result.
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  Neighbor Sum Distinguishing Total Colorings of Graphs with Bounded Maximum Average Degree

  Ai Jun DONG, Guang Hui WANG

  Acta Mathematica Sinica. 2014, 30(4): 703-709. https://doi.org/10.1007/s10114-014-2454-7

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  A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, . . . ,h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge uv ∈ E(G), if w(u) = w(v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndi (G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad(G) < 3, then tndi (G) ≤ k + 2 where k = max{Δ(G), 5}. It partially confirms the conjecture proposed by Pilśniak and Woźniak.
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  Kernel Quantile Estimator with ICI Adaptive Bandwidth Selection Technique

  Jie Yu FAN, Man Lai TANG, Mao Zai TIAN

  Acta Mathematica Sinica. 2014, 30(4): 710-722. https://doi.org/10.1007/s10114-014-1233-9

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  In this article, we consider a class of kernel quantile estimators which is the linear combination of order statistics. This class of kernel quantile estimators can be regarded as an extension of some existing estimators. The exact mean square error expression for this class of estimators will be provided when data are uniformly distributed. The implementation of these estimators depends mostly on the bandwidth selection. We then develop an adaptive method for bandwidth selection based on the intersection confidence intervals (ICI) principle. Monte Carlo studies demonstrate that our proposed approach is comparatively remarkable. We illustrate our method with a real data set.