In this paper, the authors study ground states for a class of K-component coupled nonlinear Schrödinger equations with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages three major difficulties: One is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the ℤ ^N-translation invariance, many effective methods for periodic problems cannot be applied to asymptotically periodic ones. The third difficulty is singular potential ${{{\mu _i}} \over {{{\left| x \right|}^2}}}$, which does not belong to the Kato’s class. These enable them to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential.

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in ℝ ⁿ. They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces. In particular, they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere.

By a procedure of successive projections, the authors decompose a coupled system of wave equations into a sequence of sub-systems. Then, they can clarify the indirect controls and the total number of controls. Moreover, the authors give a uniqueness theorem of solution to the system of wave equations under Kalman’s rank condition.

In this paper, the authors discuss a generalization of Lappan’s theorem to higher dimensional complex projective space and get the following result: Let f be a holomorphic mapping of Δ into ℙ ⁿ(ℂ), and let H ₁, ⋯, H _q be hyperplanes in general position in ℙ ⁿ(ℂ). Assume that $\sup \left\{ {\left( {1 - {{\left| z \right|}^2}} \right){f^\sharp }\left( z \right):z \in \bigcup\limits_{j = 1}^q {{f^{ - 1}}\left( {{H_j}} \right)} } \right\} < \infty ,$ if q ≥ 2n ² + 3, then f is normal.

Let f: ℂ → ℙ ⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.

In this paper, the authors investigate the boundedness of Toeplitz product T _f T _g and Hankel product H* _f H _g on Fock-Sobolev space for $f,g \in {\cal P}$. As a result, the boundedness of Toeplitz operator T _f and Hankel operator H _f with $f \in {\cal P}$ is characterized.

In this paper, the author partly proves a supercongruence conjectured by Z.-W. Sun in 2013. Let p be an odd prime and let a ∈ ℤ⁺. Then, if p ≡ 1 (mod 3) $\sum\limits_{k = 0}^{\left\lfloor {{5 \over 6}{p^a}} \right\rfloor } {{{\left( {\matrix{{2k} \cr k \cr } } \right)} \over {{{16}^k}}} \equiv \left( {{3 \over {{p^a}}}} \right)\,\,\left( {\bmod \,{p^2}} \right)} $ is obtained, where (÷) is the Jacobi symbol.

In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface (Σ, g) with smooth boundary ∂Σ. Explicitly, let ${\lambda _1}\left( {\partial \sum } \right) = \mathop {\inf }\limits_{u \in {W^{1,2}}\left( {\sum ,g} \right),\int_{\partial \sum } {u{\rm{d}}{s_g} = 0,u\not \equiv 0} } {{\int_\sum {\left( {{{\left| {{\nabla _g}u} \right|}^2} + {u^2}} \right){\rm{d}}{v_g}} } \over {\int_{\partial \sum } {{u^2}{\rm{d}}{s_g}} }}$ and ${\cal H} = \left\{ {u \in {W^{1,2}}\left( {\sum ,g} \right):\int_\sum {\left( {{{\left| {{\nabla _g}u} \right|}^2} + {u^2}} \right){\rm{d}}{v_g} - \alpha \int_{\partial \sum } {{u^2}{\rm{d}}{s_g} \le 1\,\,\,\,\,{\rm{and}}\,\,\,\int_{\partial \sum } {u\,{\rm{d}}{s_g} = 0} } } } \right\},$ where W ^1,2(Σ, g) denotes the usual Sobolev space and ∇ _g stands for the gradient operator. By the method of blow-up analysis, we obtain $\mathop {\sup }\limits_{u \in {\cal H}} \int_{\partial \sum } {{{\rm{e}}^{\pi {u^2}}}{\rm{d}}{s_g}} \left\{ {\matrix{{ < + \infty ,} \hfill & {0 \le \alpha < {\lambda _1}\left( {\partial \sum } \right),} \hfill \cr { = + \infty ,} \hfill & {\alpha \ge {\lambda _1}\left( {\partial \sum } \right).} \hfill \cr } } \right.$ Moreover, the author proves the above supremum is attained by a function ${u_\alpha } \in {\cal H} \cap \,{C^\infty }\left( {\overline \sum } \right)$ for any 0 ≤ α < λ₁(∂Σ). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363–386], [Yang, Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary, Pacific J. Math., 227, 2006, 177–200] and [Yang, Y., Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015, 3161–3193].

In this paper, the author computes canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

In this paper, the authors introduce a new effective method to compute the generators of the E ₁-term of the May spectral sequence. This helps them to obtain four families of non-trivial product elements in the stable homotopy groups of spheres.

In this paper, the authors give a characterization theorem for the standard tori $\mathbb{S}^1(a) \times \mathbb{S}^1(b)$, a, b > 0, as the compact Lagrangian ξ-submanifolds in the two-dimensional complex Euclidean space ℂ², and obtain the best version of a former rigidity theorem for compact Lagrangian ξ-submanifold in ℂ². Furthermore, their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in ℂ².