The Star of David Rule

In this note, a new concept called {\em $SDR$-matrix} is proposed, which is an infinite lower triangular matrix obeying the generalized rule of David star. Some basic properties of $SDR$-matrices are discussed and two conjectures on $SDR$-matrices are presented, one of which states that if a matrix is a $SDR$-matrix, then so is its matrix inverse (if exists).Comment: 7 pages, 1 figur

Conditions implying regularity of the three dimensional Navier-Stokes equation

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac like inequalities. As part of the our methods, we give a different approach to a priori estimates of Foias, Guillope and Temam.Comment: Also available at http://www.math.missouri.edu/~stephen/preprints/ (Changes: this is a substantial rewrite of the previous version.

Two kinds of hook length formulas for complete mm-ary trees

In this paper, we define two kinds of hook length for internal vertices of complete $m$-ary trees, and deduce their corresponding hook length formulas, which generalize the main results obtained by Du and Liu.Comment: 6 pages, 1 figure. To appear in Discrete Mathematic