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1. CMB 2015 (vol 58 pp. 393)

Tang, Zhongming
 On Stanley Depths of Certain Monomial Factor Algebras Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$-variables over a field $K$ and $I$ a monomial ideal of $S$. According to one standard primary decomposition of $I$, we get a Stanley decomposition of the monomial factor algebra $S/I$. Using this Stanley decomposition, one can estimate the Stanley depth of $S/I$. It is proved that ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {size}}_S(I)$. When $I$ is squarefree and ${\operatorname {bigsize}}_S(I)\leq 2$, the Stanley conjecture holds for $S/I$, i.e., ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {depth}}_S(S/I)$. Keywords:monomial ideal, size, Stanley depthCategories:13F20, 13C15

2. CMB 2014 (vol 58 pp. 105)

 On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups The prime vertex graph, $\Delta (X)$, and the common divisor graph, $\Gamma (X)$, are two graphs that have been defined on a set of positive integers $X$. Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups. Keywords:prime vertex graph, common divisor graph, character degree, class sizes, graph operationCategories:20E45, 05C25, 05C76