In this paper, the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ∂-irreducible in terms of distances between some kinds of vertex subsets of the curve complex and the arc complex of F.

For an irreducible character χ of a finite group G, the codegree of χ is defined as ∣G: ker(χ)∣/χ(1). In this paper, the authors determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, which are L₂(2 ^f) for f ≥ 2, PGL₂(q) for odd q ≥ 5 or M₁₀.

This paper concerns the even L _p Gaussian Minkowski problem in n-dimensional Euclidean space ℝ ⁿ. The existence of the solution to the even L _p Guassian Minkowski problem for p > n is obtained.

In this paper, for 1 < p < ∞, the authors show that the coarse ℓ ^p-Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu’s Property (H).

A toric origami manifold, introduced by Cannas da Silva, Guillemin and Pires, is a generalization of a toric symplectic manifold. For a toric symplectic manifold, its equivariant Chern classes can be described in terms of the corresponding Delzant polytope and the stabilization of its tangent bundle splits as a direct sum of complex line bundles. But in general a toric origami manifold is not simply connected, so the algebraic topology of a toric origami manifold is more difficult than a toric symplectic manifold. In this paper they give an explicit formula of the equivariant Chern classes of an oriented toric origami manifold in terms of the corresponding origami template. Furthermore, they prove the stabilization of the tangent bundle of an oriented toric origami manifold also splits as a direct sum of complex line bundles.

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph. Moreover, they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.

In this paper, the authors study the global regularity of the 3D magnetohydrodynamics system in terms of one velocity component. In particular, they establish a new Prodi-Serrin type regularity criterion in the framework of weak Lebesgue spaces both in time and space variables.

In this paper, the authors study the integral operator ${S_\phi }f(z) = \int_{\mathbb{C}} \phi (z,\overline w )f(w){\rm{d}}{\lambda _\alpha }(w)$

induced by a kernel function ϕ(z,·) ∈ F _α ^∞ between Fock spaces. For 1 ≤ p ≤ ∞, they prove that S _ϕ: F _α ¹ → F _α ^p is bounded if and only if $\mathop {\sup }\limits_{a \in \mathbb{C}} ||{S_\phi }{k_a}|{|_{p,\alpha }} < \infty ,$

where k _a is the normalized reproducing kernel of F _α ²; and, S _ϕ: F _α ¹ → F _α ^p is compact if and only if $\mathop {lim}\limits_{|a| \to \infty } ||{S_\phi }{k_a}|{|_{p,\alpha }} = 0.$

When 1 < q ≤ ∞, it is also proved that the condition (†) is not sufficient for boundedness of S _ϕ: F _α ^q → F _α ^p.

In the particular case $\phi (z,\overline w ) = {e^{\alpha z\overline w }}\varphi (z - \overline w )$ with φ ∈ F _α ², for 1 ≤ q < p < ∞, they show that S _ϕ: F _α ^p → F _α ^q is bounded if and only if φ = 0; for 1 < p ≤ q < ∞, they give sufficient conditions for the boundedness or compactness of the operator S _ϕ: F _α ^p → F _α ^q.

Heat exchange plays an important role in hydrodynamical systems, which is an interesting topic in theory and application. In this paper, the authors consider the global stability of steady supersonic Rayleigh flows for the one-dimensional compressible Euler equations with heat exchange, under the small perturbations of initial and boundary conditions in a finite rectilinear duct.

In this paper the author investigates the following predator-prey model with prey-taxis and rotational flux terms $\left\{ {\matrix{{{u_t} = \Delta u - \nabla \cdot (uS(x,u,v)\nabla v) + \gamma uF(v) - uh(u),} \hfill & {x \in \Omega ,\,\,\,\,\,t > 0,} \hfill \cr {{v_t} = D\Delta v - uF(v) + f(v),} \hfill & {x \in \Omega ,\,\,\,\,\,t > 0} \hfill \cr } } \right.\,\,\,\,( * )$

in a bounded domain with smooth boundary. He presents the global existence of generalized solutions to the model (*) in any dimension.