Geometric flows on warped product manifold

We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and hyperbolic geometric flow (HGF) and evolution equations on warped product manifold. We characterize the behavior of warping function under RF and under HGF. Simultaneously, we give some simple examples to illustrate the existence of such warping function solution. In addition, we also gain the evolution equations for metrics and Ricci curvature on a general warped product manifold and specific warped product manifold whose second factor manifold is of Einstein metric.Comment: 21 page

Pressure-dependent flow behavior of Zr_(41.2)Ti_(13.8)Cu_(12.5)Ni_(10)Be_(22.5) bulk metallic glass

An experimental study of the inelastic deformation of bulk metallic glass Zr_(41.2)Ti_(13.8)Cu_(12.5)Ni_(10)Be_(22.5) under multiaxial compression using a confining sleeve technique is presented. In contrast to the catastrophic shear failure (brittle) in uniaxial compression, the metallic glass exhibited large inelastic deformation of more than 10% under confinement, demonstrating the nature of ductile deformation under constrained conditions in spite of the long-range disordered characteristic of the material. It was found that the metallic glass followed a pressure (p) dependent Tresca criterion τ = τ0 + βp, and the coefficient of the pressure dependence β was 0.17. Multiple parallel shear bands oriented at 45° to the loading direction were observed on the surfaces of the deformed specimens and were responsible for the overall inelastic deformation

Trees with Maximum p-Reinforcement Number

Let $G=(V,E)$ be a graph and $p$ a positive integer. The $p$-domination number \g_p(G) is the minimum cardinality of a set $D\subseteq V$ with $|N_G(x)\cap D|\geq p$ for all $x\in V\setminus D$. The $p$-reinforcement number $r_p(G)$ is the smallest number of edges whose addition to $G$ results in a graph $G'$ with \g_p(G')<\g_p(G). Recently, it was proved by Lu et al. that $r_p(T)\leq p+1$ for a tree $T$ and $p\geq 2$. In this paper, we characterize all trees attaining this upper bound for $p\geq 3$