Using the complete lift on tangent bundles, the authors construct the complete lift on cotangent bundles of tensor fields with the aid of a musical isomorphism. In this new framework, the authors have a new intrepretation of the complete lift of tensor fields on cotangent bundles.

A small cover is a closed manifold M ⁿ with a locally standard (ℤ₂) ⁿ-action such that its orbit space is a simple convex polytope P ⁿ. Let Δ ⁿ denote an n-simplex and P(m) an m-gon. This paper gives formulas for calculating the number of D-J equivalent classes and equivariant homeomorphism classes of orientable small covers over the product space $\Delta ^{n_1 } \times \Delta ^{n_2 } \times P(m)$, where n ₁ is odd.

In this paper, the author considers a class of bounded pseudoconvex domains, i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Kähler metric g ^{Ω(μ, m)} of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Kähler-Einstein metric, the Carathéodary metric, and the Koboyashi metric are equivalent for Ω(μ, m).

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the (**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki’s question (2013), and then obtain several results of (**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the (**)-Haagerup property. Some heredity results concerning the (**)-Haagerup property are also proved.

In this paper, the authors consider the Harry-Dym equation on the line with decaying initial value. They construct the solution of the Harry-Dym equation via the solution of a 2 × 2 matrix Riemann-Hilbert problem in the complex plane. Further, one-cusp soliton solution is expressed in terms of the Riemann-Hilbert problem.

A book embedding of a graph G consists of placing the vertices of G on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is a measure of the quality of a book embedding which is the minimum number of pages in which the graph G can be embedded. In this paper, the authors discuss the embedding of the generalized Petersen graph and determine that the page number of the generalized Petersen graph is three in some situations, which is best possible.

In this paper, the authors consider the expansion problem of a wedge of gas into vacuum for the two-dimensional Euler equations in isothermal flow. By the bootstrapping argument, they prove the global existence of the smooth solution through the direct method in the case $0 < \theta \leqslant \bar \theta = \arctan \tfrac{1}{{\sqrt {2 + \sqrt 5 } }}$, where θ is the half angle of the wedge. Furthermore, they get the uniform C ^1,1 estimates of the solution to the expansion problem.

The author mainly uses the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory to study the properties of brake subharmonic solutions for the Hamiltonian systems ż(t) = J∇H(t, z(t)), where $H(t,z) = \tfrac{1}{2}(\hat B(t)z,z) + \hat H(t,z),\hat B(t)$, is a semipositive symmetric continuous matrix and Ĥ is unbounded and not uniformly coercive. It is proved that when the positive integers j and k satisfy the certain conditions, there exists a jT-periodic nonconstant brake solution zj such that zj and zkj are distinct.

The Webster scalar curvature is computed for the sphere bundle T ₁ S of a Finsler surface (S, F) subject to the Chern-Hamilton notion of adapted metrics. As an application, it is derived that in this setting (T ₁ S, g _Sasaki) is a Sasakian manifold homothetic with a generalized Berger sphere, and that a natural Cartan structure is arising from the horizontal 1-forms and the author associates a non-Einstein pseudo-Hermitian structure. Also, one studies when the Sasaki type metric of T ₁ S is generally adapted to the natural co-frame provided by the Finsler structure.

First, the authors give a Gröbner-Shirshov basis of the finite-dimensional irreducible module V _q(λ) of the Drinfeld-Jimbo quantum group U _q(G ₂) by using the double free module method and the known Gröbner-Shirshov basis of Uq(G2). Then, by specializing a suitable version of U _q(G ₂) at q = 1, they get a Gröbner-Shirshov basis of the universal enveloping algebra U(G ₂) of the simple Lie algebra of type G ₂ and the finite-dimensional irreducible U(G ₂)-module V (λ).

In this paper, the Riemann problem with delta initial data for the one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics is considered. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtained the global existence of generalized solutions which contains delta-shock. Moreover, the author obtains the stability of generalized solutions by making use of the perturbation of the initial data.

The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained.

Consider the initial boundary value problem of the strong degenerate parabolic equation $\begin{array}{*{20}c} {\partial _{xx} u + u\partial _y u - \partial _t u = f(x,y,t,u),} & {(x,y,t) \in Q_T = \Omega \times (0,T)} \\ \end{array}$ with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff’s theorem, the solvability of the equation is obtained in BV (Q _T) sense. The stability of the solutions is obtained by Kruzkov’s double variables method.