Shan Liu — Page 7 | Searcharxiv

arXiv2011

Effect of a crossing change on crossing number

The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the link after crossing change can be estimated when certain conditions are met. At the end of the article, we give an example to demonstrate a special case for the theorem and a counterexample to explain that the theorem cannot be applied if the obtained link is not alternating.

math.GT↗

arXiv2006

On the role of confinement on solidification in pure materials and binary alloys

We use a phase-field model to study the effect of confinement on dendritic growth, in a pure material solidifying in an undercooled melt, and in the directional solidification of a dilute binary alloy. Specifically, we observe the effect of varying the vertical domain extent ($δ$) on tip selection, by quantifying the dendrite tip velocity and curvature as a function of $δ$, and other process parameters. As $δ$ decreases, we find that the operating state of the dendrite tips becomes significantly affected by the presence of finite boundaries. For particular boundary conditions, we observe a switching of the growth state from 3-D to 2-D at very small $δ$, in both the pure material and alloy. We demonstrate that results from the alloy model compare favorably with those from an experimental study investigating this effect.

cond-mat.mtrl-sci↗