Authors:Yan Tan; Zhi Xiang Wu Pages: 933 - 946 Abstract: In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-7065-7 Issue No:Vol. 34, No. 6 (2018)
Authors:Jaume Llibre; Clàudia Valls Pages: 947 - 958 Abstract: We study the Hindmarsh–Rose burster which can be described by the differential system $$\dot x = y - {x^3} + b{x^2} + I - z,\dot y = 1 - 5{x^2} - y,\dot z = \mu \left( {s\left( {x - {x_0}} \right) - z} \right),$$ where b, I, μ, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-5661-1 Issue No:Vol. 34, No. 6 (2018)
Authors:Yang Cao; Jing Xue Yin; Ying Hua Li Pages: 959 - 974 Abstract: In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ(s) and sources A(s). We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ(s) and A(s) are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T > 0, the solution will never blow up during the period 0 < t < T, so long as the viscous coefficient is large enough. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-7245-5 Issue No:Vol. 34, No. 6 (2018)
Authors:Aung Zaw Myint; Li Li; Ming Xin Wang Pages: 975 - 991 Abstract: This paper deals with one kind of Belousov–Zhabotinskii reaction model. Linear stability is discussed for the spatially homogeneous problem firstly. Then we focus on the stationary problem with diffusion. Non-existence and existence of non-constant positive solutions are obtained by using implicit function theorem and Leray–Schauder degree theory, respectively. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-7295-8 Issue No:Vol. 34, No. 6 (2018)
Authors:Wei Hua Wang; Gang Wu Pages: 992 - 1000 Abstract: In this paper, we establish the global well-posedness of the generalized rotating magnetohydrodynamics equations if the initial data are in X1−2α defined by \({x^{1 - 2\alpha }} = \left\{ {u \in D'\left( {{R^3}} \right):{{\int_{{R^3}} {\left \xi \right } }^{1 - 2\alpha }}\left {\hat u\left( \xi \right)} \right d\xi < + \infty } \right\}\) . In addition, we also give Gevrey class regularity of the solution. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-7276-y Issue No:Vol. 34, No. 6 (2018)
Authors:Rui Li; Bei Liu; Rui Liu; Qing Yue Zhang Pages: 1001 - 1014 Abstract: The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1). We first show that the shifts ϕ(· − k) (k ∈ ℤd+1) are Lp,q-stable if and only if for any ξ ∈ ℝd+1, \(\sum\nolimits_{k \in \mathbb{Z}^{d + 1} } {\left {\hat \varphi (\xi + 2\pi k)} \right ^2 > 0}\) . Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1) to be Lp,q-stable which improves some known results. PubDate: 2018-06-01 DOI: 10.1007/s10114-018-7333-1 Issue No:Vol. 34, No. 6 (2018)
Authors:Xin Jun Gao Pages: 1015 - 1027 Abstract: We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg–de Vries equation in Sobolev spaces H s (ℝ) for s > \( - \frac{3}{{22}}\) . The main approach is the “I-method” together with the multilinear multiplier analysis. PubDate: 2018-06-01 DOI: 10.1007/s10114-018-7241-4 Issue No:Vol. 34, No. 6 (2018)
Authors:Chao Lu; Jing Lu Pages: 1028 - 1036 Abstract: In this paper, we study the unconditional uniqueness of solution for the Cauchy problem of \(\dot H^{s_c } (0 \leqslant s_c < 2)\) critical nonlinear fourth-order Schrödinger equations i∂ t u+Δ2u−ϵu = λ u α u. By employing paraproduct estimates and Strichartz estimates, we prove that unconditional uniqueness of solution holds in \(C_t (I;\dot H^{s_c } (\mathbb{R}^d ))\) for d ≥ 11 and \(\min \left\{ {1^ - ,\tfrac{8} {{d - 4}}} \right\} \geqslant \alpha > \frac{{ - (d - 4) + \sqrt {(d - 4)^2 + 64} }} {4}\) . PubDate: 2018-06-01 DOI: 10.1007/s10114-017-7354-1 Issue No:Vol. 34, No. 6 (2018)
Authors:Wen Peng Zhang; Xin Lin Pages: 1037 - 1049 Abstract: The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q (a positive odd number), and give an exact computational formula for it. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-7188-x Issue No:Vol. 34, No. 6 (2018)
Authors:Jin Jiang Li; Pan Wang Wang; Min Zhang Pages: 1050 - 1058 Abstract: Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) = Ʃn≤xa(n). In this paper, we establish an asymptotic formula of the fourth power moment of A(x) and prove that \(\int_1^T {{A^4}\left( x \right)dx = \frac{3}{{64\kappa {\pi ^4}}}{s_{4;2}}\left( {\tilde a} \right){T^{2\kappa }} + O\left( {{T^{2\kappa - {\delta _4} + \varepsilon }}} \right)} \) with δ4 = 1/8, which improves the previous result. PubDate: 2018-06-01 DOI: 10.1007/s10114-017-6508-5 Issue No:Vol. 34, No. 6 (2018)
Authors:Lin Wang; Xin Sheng Wang; Yu Jun Zhu Abstract: In this paper we investigate the quasi-shadowing property for C1 random dynamical systems on their random partially hyperbolic sets. It is shown that for any pseudo orbit {x k } −∞ +∞ on a random partially hyperbolic set there exists a “center” pseudo orbit {y k } −∞ +∞ shadowing it in the sense that yk+1 is obtained from the image of y k by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above “center” pseudo orbit {y k } −∞ +∞ can be chosen such that yk+1 and the image of y k lie in their common center leaf. PubDate: 2018-06-06 DOI: 10.1007/s10114-018-7314-4
Authors:Gang Yang; Xuan Shang Da Abstract: In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each X m is a Ding projective R-module for all m ∈ ℤ. PubDate: 2018-05-31 DOI: 10.1007/s10114-018-7461-7
Authors:Hendrik Herrmann; Chin Yu Hsiao; Xiao Shan Li Abstract: In this paper, we give an explicit formula for the Szegő kernel for (0, q) forms on the Heisenberg group Hn+1. PubDate: 2018-05-31 DOI: 10.1007/s10114-018-7324-2
Authors:Lin Li; Alatancang Chen; De Yu Wu Abstract: Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about symplectic self-adjointness are shown. PubDate: 2018-05-31 DOI: 10.1007/s10114-018-7267-7
Authors:Hee Chul Pak; Young Ja Park Abstract: We introduce a new function space that is much finer than the classical Lebesgue spaces, and investigate the continuity and the discontinuity of the Calderón–Zygmund operators on the new weighted spaces. In order to identify the continuity of singular integrals, we prove an interpolation theorem on the new spaces and propose a concept of the spectrum of base functions. PubDate: 2018-05-28 DOI: 10.1007/s10114-018-7043-8
Authors:Ming Ming Cao; Qing Ying Xue Abstract: The main result of this paper is a bi-parameter Tb theorem for Littlewood–Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μ n × μ m , where the measures μ n and μ m are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again. PubDate: 2018-05-23 DOI: 10.1007/s10114-018-7431-0
Authors:Yong Hu; Lei Zhang Abstract: Let S be a smooth minimal projective surface of general type with p g (S) = q(S) = 1, K S 2 = 6. We prove that the degree of the bicanonical map of S is 1 or 2. So if S has non-birational bicanonical map, then it is a double cover over either a rational surface or a K3 surface. PubDate: 2018-05-18 DOI: 10.1007/s10114-018-7262-z
Authors:Yue Liang Liang; Xiao Chun Fang Abstract: The paper is devoted to the study of generalized inductive limit of C*-algebras with coherent maps being completely positive contractions of order zero: the nuclear dimension of generalized inductive limit of C*-algebras with finite nuclear dimension is finite; the generalized inductive limits of C*-algebras with the α-comparison property also have the α-comparison property. PubDate: 2018-05-18 DOI: 10.1007/s10114-018-7336-y
Authors:Xiang Qian Li; Ru Hong Hu; Zi Hong Tian Abstract: In this article, we establish the existence of an LHMTS(m v ) for v ≡ 2 (mod 6) and m ≡ 3 (mod 6). Thus there exists an LHMTS(m v ) if and only if v(v − 1)m2 ≡ 0 (mod 3) except possibly for v = 6, m ≡ 1,5 (mod 6) and m ≠ 1. In the similar way, the existence of LHDTS(m v ) is completely determined, i.e., there exists an LHDTS(m v ) if and only if v(v − 1)m2 ≡ 0 (mod 3). PubDate: 2018-05-18 DOI: 10.1007/s10114-018-7306-4
Authors:Xiang Yu Zhou; Lang Feng Zhu Abstract: In this paper, we discuss our most recent results on the optimal L2 extension problem and Siu’s lemma as applications of the strong openness property of multiplier ideal sheaves obtained by Guan and Zhou. PubDate: 2018-05-18 DOI: 10.1007/s10114-018-7539-2
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