Let (M,ω) be a symplectic manifold. In this paper, the authors consider the notions of musical (bemolle and diesis) isomorphisms ω ^b: TM → T*M and ω ^#: T*M → TM between tangent and cotangent bundles. The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω ^b-related. As consequence of analyze of connections between the complete lift ^c ω _TM of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle, the authors proved that dp is a pullback of ^c ω _TM by ω ^#. Also, the authors investigate the complete lift ^cϕ _T*M of almost complex structure ϕ to cotangent bundle and prove that it is a transform by of complete lift ^c ϕ _TM to tangent bundle if the triple (M, ω,ϕ) is an almost holomorphic $<mi xmlns:xlink="http://www.w3.org/1999/xlink" mathvariant="fraktur">A</mi>$-manifold. The transform of complete lifts of vector-valued 2-form is also studied.

Let f be a nonconstant meromorphic function, c ∊ ℂ, and let a(z)(≢ 0) ∊ S(f) be a meromorphic function. If f(z) and P(z, f(z)) share the sets {a(z), −a(z)}, {0} CM almost and share {∞} IM almost, where P(z, f(z)) is defined as (1.1), then f(z) ≡ ±P(z, f(z)) or f(z)P(z, f(z)) ≡ ±a ²(z). This extends the results due to Chen and Chen (2013), Liu (2009) and Yi (1987).

In this paper the author studies the initial boundary value problem of semilinear wave systems in exterior domain in high dimensions (n ≥ 3). Blow up result is established and what is more, the author gets the upper bound of the lifespan. For this purpose the test function method is used.

Given ϕ a subharmonic function on the complex plane ℂ, with ΔϕdA being a doubling measure, the author studies Fock Carleson measures and some characterizations on μ such that the induced positive Toeplitz operator T _μ is bounded or compact between the doubling Fock space $F_\phi ^p$ and $F_\phi ^\infty $ with 0 < p ≤ ∞, where μ is a positive Borel measure on ℂ.

In this paper, the authors give the local L ² estimate of the maximal operator$S_{\phi ,\gamma }^ * $ of the operator family {S _{t,ϕ, γ}} defined initially by${S_{t,\phi ,\gamma }}f(x): = {{\rm{e}}^{{\rm{i}}\,t\phi (\sqrt { - \Delta } )}}f(\gamma (x,t)) = {(2\pi )^{ - 1}}\int_\mathbb{R} {{{\rm{e}}^{{\rm{i}}\gamma (x,t) \cdot \xi + {\rm{i}}\,t\phi ({\rm{|}}\xi {\rm{|}})}}} \hat f(\xi ){\rm{d}}\xi ,\;\;\;\;\;\;\;\;f \in {\cal S}(\mathbb{R}),$ which is the solution (when {itn} = 1) of the following dispersive equations (*) along a curve {itγ}: $\left\{ {\matrix{ {{\rm{i}}{\partial _t}u + \phi (\sqrt { - {\rm{\Delta }}} )u = 0,} \hfill \;\;\;\;\; {(x,t) \in \mathbb{R}{^n} \times \mathbb{R},} \hfill \cr {u(x,0) = f(x),} \hfill \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {f \in {\cal S}({\mathbb{R}^n}),} \hfill \cr } } \right.$ where {itϕ}: ℝ⁺ → ℝ satisfies some suitable conditions and$\phi (\sqrt { - {\rm{\Delta }}} )$ is a pseudo-differential operator with symbol {itϕ}(∣{itξ}∣). As a consequence of the above result, the authors give the pointwise convergence of the solution (when {itn} = 1) of the equation (*) along curve {itγ}. Moreover, a global {itL}² estimate of the maximal operator$S_{\phi ,\gamma }^ * $ is also given in this paper.

In this paper, the authors prove a Nekhoroshev type theorem for the nonlinear wave equation ${u_{tt}} = {u_{xx}} - mu - f(u),\;\;\;\;\;x \in [0,\pi ]$ in Gevrey space.

Win proved a well-known result that the graph G of connectivity κ(G) with α(G) ≤ κ(G) + k − 1 (k ≥ 2) has a spanning k-ended tree, i.e., a spanning tree with at most k leaves. In this paper, the authors extended the Win theorem in case when κ(G) = 1 to the following: Let G be a simple connected graph of order large enough such that α(G) ≤ k + 1 (k ≥ 3) and such that the number of maximum independent sets of cardinality k + 1 is at most n − 2k − 2. Then G has a spanning k-ended tree.

In this paper, the author first introduces the concept of generalized algebraic cone metric spaces and some elementary results concerning generalized algebraic cone metric spaces. Next, by using these results, some new fixed point theorems on generalized (complete) algebraic cone metric spaces are proved and an example is given. As a consequence, the main results generalize the corresponding results in complete algebraic cone metric spaces and generalized complete metric spaces.

In the present paper, the rigidity of hypersurfaces in Euclidean space is revisited. The Darboux equation is highlighted and two new proofs of the rigidity are given via energy method and maximal principle, respectively.

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.

In this paper, the authors consider the problem of which (generalized) moment-angle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope P _v, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to P _v also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.