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 ∈ C^γ (T¹) with γ > 1 − 1/n and for each j with 0 ≤ j ≤ n, P _j ∈ L(T¹) where T¹ = R/Z.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.