In this paper, the author studies the multidimensional stability of subsonic phase transitions in a steady supersonic flow of van der Waals type. The viscosity capillarity criterion (in “Arch. Rat. Mech. Anal., 81(4), 1983, 301–315”) is used to seek physical admissible planar waves. By showing the Lopatinski determinant being non-zero, it is proved that subsonic phase transitions are uniformly stable in the sense of Majda (in “Mem. Amer. Math. Soc., 41(275), 1983, 1–95”) under both one dimensional and multidimensional perturbations.

The authors prove that every complex Banach space admits an equivalent real norm that is far away from being a complex norm. Furthermore, this real norm can be chosen to share many properties with complex norms, but it is still not a complex norm.

A 2-coupled nonlinear Schrödinger equations with bounded varying potentials and strongly attractive interactions is considered. When the attractive interaction is strong enough, the existence of a ground state for sufficiently small Planck constant is proved. As the Planck constant approaches zero, it is proved that one of the components concentrates at a minimum point of the ground state energy function which is defined in Section 4.

Motivated by the results of J. Y. Chemin in “J. Anal. Math., 77, 1999, 27–50” and G. Furioli et al in “Revista Mat. Iberoamer., 16, 2002, 605–667”, the author considers further regularities of the mild solutions to Navier-Stokes equation with initial data u ₀ ∈ L ^d(ℝ ^d). In particular, it is proved that of u ∈ C([0, T*); L ^d(ℝ ^d)) is a mild solution of (N S _v), then $u(t,x) - e^{\nu t\Delta } u_0 \in \tilde L^\infty ((0,T);\dot B_{\frac{d}{2},\infty }^1 ) \cap \tilde L^1 ((0,T);\dot B_{\frac{d}{2},\infty }^3 )$ for any T < T*.

Based on a differentiable merit function proposed by Taji et al. in “Math. Prog. Stud., 58, 1993, 369–383”, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors from an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π _* S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres $\pi _{p^n q + 2pq + q - 3} S$ which is of order p and is represented by k ₀ h _n ∈ $Ext_A^{3,p^n q + 2pq + q} $(ℤ _{p, ℤp}) in the Adams spectral sequence, where p ≥ 5 is an odd prime, n ≥ 3 and q = 2(p − 1). In the course of the proof, a new family of homotopy elements in $\pi _{p^n q + (p + 1)q - 1} V(1)$ which is represented by β _* i′_* i _*(h _n) ∈ $Ext_A^{2,p^n q + (p + 1)q + 1} $(H ^* V(1), ℤ _p) in the Adams sequence is detected.

In this paper, the partial positivity (resp., negativity) of the curvature of all irreducible Riemannian symmetric spaces is determined. From the classifications of abstract root systems and maximal subsystems, the author gives the calculations for symmetric spaces both in classical types and in exceptional types.

In this paper, it is proved that any self-affine set satisfying the strong separation condition is uniformly porous. The author constructs a self-affine set which is not porous, although the open set condition holds. Besides, the author also gives a C ¹ iterated function system such that its invariant set is not porous.