The Hasse-Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse-Weil bound for the curve defined by y − y = f(x) over the fini...

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

We study upward planar straight-line embeddings (UPSE) of directed trees on given point sets. The set S has size at least the number vertices in tree. For special case where tree is a path P we show that: (a) If one-sided convex, UPSE s equals maximal monotone paths P. (b) general position and composed by three paths, middle longer than other two, then it always admits an S. that decision probl...

This study examined the probability of (a) being placed in a disciplinary alternative education setting for mandatory versus discretionary reasons and (b) returning within the same year among an ethnically diverse sample (African American, Caucasian, Hispanic) of middle and high school students (N=270). Participants were compared based on ethnicity, gender, grade level, and special education st...