This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.

Let $\mathcal{R}_\mathcal{III}(n)$ be the classical domain of type $\mathcal{III}$ with n ≥ 2. This article is devoted to a deep study of the Schwarz lemma on $\mathcal{R}_\mathcal{III}(n)$ via not only exploring the smooth boundary points of $\mathcal{R}_\mathcal{III}(n)$) but also proving the Schwarz lemma at the smooth boundary point for holomorphic self-mappings of $\mathcal{R}_\mathcal{III}(n)$.

A classical problem of D. H. Lehmer suggests the study of distributions of elements of Z/p Z of opposite parity to the multiplicative inverse mod p. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved, from which one may see that Zhang’s error term is optimal up to the logarithm factor. The method also applies to the case of arbitrary positive integral moments.

The one-dimensional compressible non-Newtonian models are considered in this paper. The extra-stress tensor in our models satisfies a kind of power law structure which was proposed by O. A. Ladyzhenskaya in 1970s. In particular, the viscosity coefficient in our models depends on the density. By using energy-estimate, the authors obtain the existence and uniqueness of local strong solutions for which the density is non-negative.

This paper deals with the Briot-Bouquet differential equations with degree three. The previous result shows that all the meromorphic solutions belong to W. Here, by applying the Kowalevski-Gambier method, the authors give all the possible explicit meromorphic solutions. The result is more applicable. Also, this method can be used to deal with the more general Briot-Bouquet differential equations.

In this paper, the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation ${\Delta _{p,f}}u + c{u^\sigma } = 0$ on noncompact smooth metric measure space, where c is a nonnegative constant, and p, σ (1 < p ≤ 2, σ ≤ p - 1) are real constants. Moreover, by the gradient estimate, they can get the corresponding Liouville theorem and Harnack inequality.

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.

In this paper, the author establishes a reduction theorem for linear Schrödinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM (Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schrödinger operator possesses the property of pure point spectra and zero Lyapunov exponent.

The author obtains that the asymptotic relations $\mathbb{P}\left( {\sum\limits_{i = 1}^n {{\theta _i}{X_i}} >x} \right) \sim \mathbb{P}\left( {\mathop {\max }\limits_{1 \le m \le n} \sum\limits_{i = 1}^m {{\theta _i}{X_i}} >x} \right) \sim \mathbb{P}\left( {\mathop {\max {\theta _i}{X_i}}\limits_{1 \le i \le n} >x} \right) \sim \sum\limits_{i = 1}^n \mathbb{P}{\left( {{\theta _i}{X_i} >x} \right)}$ hold as x → ∞, where the random weights θ ₁,..., θ _n are bounded away both from 0 and from ∞ with no dependency assumptions, independent of the primary random variables X ₁,..., X _n which have a certain kind of dependence structure and follow non-identically subexponential distributions. In particular, the asymptotic relations remain true when X ₁,..., X _n jointly follow a pairwise Sarmanov distribution.

In this paper, the author establishs a real-valued function on Kähler manifolds by holomorphic sectional curvature under parallel translation. The author proves if such functions are equal for two simply-connected, complete Kähler manifolds, then they are holomorphically isometric.

In this paper, the authors present a method to construct the minimal and H-minimal Lagrangian submanifolds in complex hyperquadric Q _n from submanifolds with special properties in odd-dimensional spheres. The authors also provide some detailed examples.

This paper is devoted to the L ^p (p > 1) solutions of one-dimensional backward stochastic differential equations (BSDEs for short) with general time intervals and generators satisfying some non-uniform conditions in t and ω. An existence and uniqueness result, a comparison theorem and an existence result for the minimal solutions are respectively obtained, which considerably improve some known works. Some classical techniques used to deal with the existence and uniqueness of L ^p (p > 1) solutions of BSDEs with Lipschitz or linear-growth generators are also developed in this paper.