In this paper, a model is derived to describe a quartic anharmonic interatomic interaction with an external potential involving a pair electron-phonon. The authors study the corresponding Cauchy Problem in the semilinear and quasilinear cases.

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n(T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.

In this paper, the Laplacian on the holomorphic tangent bundle T ^1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M, F). Utilizing the initiated “Bochner technique”, a vanishing theorem for vector fields on the holomorphic tangent bundle T ^1,0 M is obtained.

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the following Cauchy problem: $\left\{ \begin{gathered} \square u = \left( {u_t } \right)^3 , n = 2, \hfill \\ t = 0: u = 0, u_t = \varepsilon g\left( x \right), x \in \mathbb{R}^2 , \hfill \\\end{gathered} \right.$ where $\square = \partial _t^2 - \sum\limits_{i = 1}^n {\partial _{x_i }^2 }$ is the wave operator, g(x) ≢ 0 is a smooth non-negative function on ℝ² with compact support, and ɛ > 0 is a small parameter. It is shown that the solution blows up in a finite time, and the lifespan T(ɛ) of solutions has an upper bound T(ɛ) ≤ exp(Aɛ ⁻²) with a positive constant A independent of ɛ, which belongs to the same kind of the lower bound of the lifespan.

For any given coprime integers p and q greater than 1, in 1959, B. J. Birch proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p ^a q ^b. As Davenport observed, Birch’s proof can be modified to show that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari gave an effective version of this bound. The author improves this bound.

There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.

This paper is devoted to the analysis of the Cauchy problem for a system of PDEs arising in radiative hydrodynamics. This system, which comes from the so-called equilibrium diffusion regime, is a variant of the usual Euler equations, where the energy and pressure functionals are modified to take into account the effect of radiation and the energy balance containing a nonlinear diffusion term acting on the temperature. The problem is studied in the multi-dimensional framework. The authors identify the existence of a strictly convex entropy and a stability property of the system, and check that the Kawashima-Shizuta condition holds. Then, based on these structure properties, the well-posedness close to a constant state can be proved by using fine energy estimates. The asymptotic decay of the solutions are also investigated.

The authors consider the complex Monge-Ampère equation det\left( {u_{i\bar j} } \right) = ψ(z, u, ∇ u) in bounded strictly pseudoconvex domains Ω, subject to the singular boundary condition u = ∞ on ∂Ω. Under suitable conditions on ψ, the existence, uniqueness and the exact asymptotic behavior of solutions to boundary blow-up problems for the complex Monge-Ampère equations are established.

The authors take all isomorphism classes of indecomposable representations as new generators, and obtain all skew-commutators between these generators by using the Ringel-Hall algebra method. Then they prove that the set of these skew-commutators is a Gröbner-Shirshov basis for quantum group of type \mathbb{D}_4.

The author studies the metric spaces with operator norm localization property. It is proved that the operator norm localization property is coarsely invariant and is preserved under certain infinite union. In the case of finitely generated groups, the operator norm localization property is also preserved under the direct limits.

Herman constructed an autonomous system of two degrees of freedom which says that in non-convex situations, oscillations do happen and Aubry-Mather Theory cannot apply (see the results due to W. F. Chen in 1992). In this paper, it is shown that although the orbits could visit a region far away from the initial point in phase space, they can only exist in some fixed regions in I = (I ₁, I ₂) plane. Moreover, Aubry-Mather Theory can be applied outside the regions.

This paper presents a definition of residue formulas for the Euler class of cohomology-oriented sphere fibrations ζ. If the base of ζ is a topological manifold, a Hopf index theorem can be obtained and, for the smooth category, a generalization of a residue formula is derived for real vector bundles given in [2].

The authors study the existence of nontrivial solutions to p-Laplacian variational inclusion systems \left\{ \begin{gathered} - \Delta _p u + \left| u \right|^{p - 2} u \in \partial _1 F\left( {u,v} \right), in \mathbb{R}^N , \hfill \\ - \Delta _p v + \left| v \right|^{p - 2} v \in \partial _2 F\left( {u,v} \right), in \mathbb{R}^N , \hfill \\\end{gathered} \right. where N ≥ 2, 2 ≤ p ≤ N and F: ℝ² → ℝ is a locally Lipschitz function. Under some growth conditions on F, and by Mountain Pass Theorem and the principle of symmetric criticality, the existence of such solutions is guaranteed.

Let π be an irreducible unitary cuspidal representation of GL_m $\left( {\mathbb{A}_\mathbb{Q} } \right)$, m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.