Authors:Xiaoping Yuan Pages: 1037 - 1046 Abstract: Abstract It is shown that all solutions are bounded for Duffing equation \(\ddot x + {x^{2n + 1}} + \sum\limits_{j = 0}^{2n} {{P_j}} \left( t \right){x^j} = 0\) , provided that for each n + 1 ≤ j ≤ 2n, P j ∈ Cγ (T1) with γ > 1 − 1/n and for each j with 0 ≤ j ≤ n, P j ∈ L(T1) where T1 = R/Z. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1020-x Issue No:Vol. 38, No. 5 (2017)

Authors:Yan Guan Pages: 1047 - 1058 Abstract: Abstract In this paper the modeling of a thin plate in unilateral contact with a rigid plane is properly justified. Starting from the three-dimensional nonlinear Signorini problem, by an asymptotic approach the convergence of the displacement field as the thickness of the plate goes to zero is studied. It is shown that the transverse mechanical displacement field decouples from the in-plane components and solves an obstacle problem. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1021-9 Issue No:Vol. 38, No. 5 (2017)

Authors:Chao Gong; Yong Lin Pages: 1059 - 1070 Abstract: Abstract The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,1) and CD(K,n). PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1022-8 Issue No:Vol. 38, No. 5 (2017)

Authors:Xiangzhi Cao Pages: 1071 - 1076 Abstract: Abstract This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold \(\overline M \) of non-negative (n−1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L q -energy from complete submanifold in a partially non-negatively curved manifold to non-positively curved manifold is also obtained. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1023-7 Issue No:Vol. 38, No. 5 (2017)

Authors:Ghulam Mustafa Pages: 1077 - 1092 Abstract: Abstract The objective of this article is to introduce a generalized algorithm to produce the m-point n-ary approximating subdivision schemes (for any integer m, n ≥ 2). The proposed algorithm has been derived from uniform B-spline blending functions. In particular, we study statistical and geometrical/traditional methods for the model selection and assessment for selecting a subdivision curve from the proposed family of schemes to model noisy and noisy free data. Moreover, we also discuss the deviation of subdivision curves generated by proposed family of schemes from convex polygonal curve. Furthermore, visual performances of the schemes have been presented to compare numerically the Gibbs oscillations with the existing family of schemes. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1024-6 Issue No:Vol. 38, No. 5 (2017)

Authors:Zhihong Wen; Guantie Deng; Cuiqiao Wang; Feifei Qu Pages: 1093 - 1110 Abstract: Abstract This paper deals with two topics mentioned in the title. First, it is proved that function f in L p (∂D a ) can be decomposed into a sum g + h, where D a is an angular domain in the complex plane, g and h are the non-tangential limits of functions in H p (D a ) and \({H^p}\left( {\overline D _a^c} \right)\) in the sense of L p (D a ), respectively. Second, the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1025-5 Issue No:Vol. 38, No. 5 (2017)

Authors:Qi Yan; Jiandong Yin; Tao Wang Pages: 1111 - 1118 Abstract: Abstract In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1026-4 Issue No:Vol. 38, No. 5 (2017)

Authors:Heman Fu; Feng Tan Pages: 1119 - 1130 Abstract: Abstract For each real number λ ∈ 2 [0, 1], λ-power distributional chaos has been introduced and studied via Furstenberg families recently. The chaoticity gets stronger and stronger as λ varies from 1 to 0, where 1-power distributional chaos is exactly the usual distributional chaos. As a generalization of distributional n-chaos, λ-power distributional n-chaos is defined similarly. Lots of classic results on distributional chaos can be improved to be the versions of λ-power distributional n-chaos accordingly. A practical method for distinguishing 0-power distributional n-chaos is given. A transitive system is constructed to be 0-power distributionally n-chaotic but without any distributionally (n + 1)-scrambled tuples. For each λ ∈ 2 [0, 1], λ-power distributional n-chaos can still appear in minimal systems with zero topological entropy. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1027-3 Issue No:Vol. 38, No. 5 (2017)

Authors:Tongzhu Li Pages: 1131 - 1144 Abstract: Abstract Let x: M n → S n+1 be an immersed hypersurface in the (n + 1)-dimensional sphere S n+1. If, for any points p, q ∈ M n , there exists a Möbius transformation ϕ: S n+1 → S n+1 such that ϕ ○ x(M n ) = x(M n ) and ϕ ○ x(p) = x(q), then the hypersurface is called a Möbius homogeneous hypersurface. In this paper, the Möbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Möbius transformation. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1028-2 Issue No:Vol. 38, No. 5 (2017)

Authors:Lifeng Xi; Wen Wu; Ying Xiong Pages: 1145 - 1178 Abstract: Abstract In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R n generated from an initial cube pattern with an (n−m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by “multi-rules” take the value in Marstrand’s theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μV is absolutely continuous with respect to the Lebesgue measure L m . When μV ≪ L m , the connection of the local dimension of μV and the box dimension of slices is given. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1029-1 Issue No:Vol. 38, No. 5 (2017)

Authors:Shimin Li; Yulin Zhao Pages: 1179 - 1196 Abstract: Abstract This paper deals with the qualitative behavior of orbits at degenerate singular point with the method of quasi normal sector, which is a generalization of Frommer’s normal sectors. Several examples show that this method is more effective than the well-known methods of Z-sectors, normal sectors and generalized normal sector. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1030-8 Issue No:Vol. 38, No. 5 (2017)

Authors:Huai-Liang Chang; Jun Li; Wei-Ping Li; Chiu-Chu Melissa Liu Pages: 869 - 882 Abstract: Abstract The mixed spin P-fields (MSP for short) theory sets up a geometric platform to relate Gromov-Witten invariants of the quintic three-fold and Fan-Jarvis-Ruan-Witten invariants of the quintic polynomial in five variables. It starts with Wittens vision and the P-fields treatment of GW invariants and FJRW invariants. Then it brie y discusses the master space technique and its application to the set-up of the MSP moduli. Some key results in MSP theory are explained and some examples are provided. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1101-x Issue No:Vol. 38, No. 4 (2017)

Authors:Jinwon Choi; Young-Hoon Kiem Pages: 883 - 900 Abstract: Abstract Landau-Ginzburg/Calabi-Yau correspondence claims the equivalence between the Gromov-Witten theory and the Fan-Jarvis-Ruan-Witten theory. The authors survey recently developed wall-crossing approach to the Landau-Ginzburg/Calabi-Yau correspondence for a quintic threefold. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1102-9 Issue No:Vol. 38, No. 4 (2017)

Authors:Richard Eager; Kentaro Hori; Johanna Knapp; Mauricio Romo Pages: 901 - 912 Abstract: Abstract The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1103-8 Issue No:Vol. 38, No. 4 (2017)

Authors:Huijun Fan; Tyler Jarvis; Yongbin Ruan Pages: 913 - 936 Abstract: Abstract This is a survey article for the mathematical theory of Witten’s Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper the authors focus on the description of the moduli. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1104-7 Issue No:Vol. 38, No. 4 (2017)

Authors:Mauricio Romo Pages: 937 - 962 Abstract: Abstract The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. The author focuses on the case of G = SL(N, C) and M being a knot complement: M = S 3 \ K. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G = SL(N, C). He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1105-6 Issue No:Vol. 38, No. 4 (2017)

Authors:Yongbin Ruan Pages: 963 - 984 Abstract: Abstract The gauged linear sigma model (GLSM for short) is a 2d quantum field theory introduced by Witten twenty years ago. Since then, it has been investigated extensively in physics by Hori and others. Recently, an algebro-geometric theory (for both abelian and nonabelian GLSMs) was developed by the author and his collaborators so that he can start to rigorously compute its invariants and check against physical predications. The abelian GLSM was relatively better understood and is the focus of current mathematical investigation. In this article, the author would like to look over the horizon and consider the nonabelian GLSM. The nonabelian case possesses some new features unavailable to the abelian GLSM. To aid the future mathematical development, the author surveys some of the key problems inspired by physics in the nonabelian GLSM. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1106-5 Issue No:Vol. 38, No. 4 (2017)

Authors:Aydin Gezer; Abdullah Magden Pages: 985 - 998 Abstract: Abstract Let (M, g) be an n-dimensional Riemannian manifold and T 2 M be its second-order tangent bundle equipped with a lift metric \(\tilde g\) . In this paper, first, the authors construct some Riemannian almost product structures on (T 2 M, \(\tilde g\) ) and present some results concerning these structures. Then, they investigate the curvature properties of (T 2 M, \(\tilde g\) ). Finally, they study the properties of two metric connections with nonvanishing torsion on (T 2 M, \(\tilde g\) ): The H-lift of the Levi-Civita connection of g to T 2 M, and the product conjugate connection defined by the Levi-Civita connection of \(\tilde g\) and an almost product structure. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1107-4 Issue No:Vol. 38, No. 4 (2017)

Authors:Yu Lu; Shenglin Zhu Pages: 999 - 1018 Abstract: Abstract In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/A H a right H*-Galois extension. The authors prove that, if A H is a separable k-algebra, then for any right coideal subalgebra B of H, the B-invariants A B = {a ∈ A b · a = ε(b)a, ∀b ∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing A H as in the classical case. The results are applied to the case H = (kG)* for a finite group G to get a Galois 1-1 correspondence. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1108-3 Issue No:Vol. 38, No. 4 (2017)