The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation $F$, has been widely studied extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie $\{0,1,\dots,r-1\}$. combinatorics generalized less well-studied. In paper, we provide exact bounds fo...

This paper gives a condition for an embedded minimal disk to be bi-Lipschitz to a piece of a helicoid. Namely, if such a disk is inside a ball (with boundary on the ball) and has large curvature at the center relative to nearby points, then, in a smaller ball, it is bi-Lipschitz to a piece of a helicoid. Moreover, the Lipschitz constant can be chosen as close to 1 as desired. This is a sharpeni...

The classical n-logarithm is a multivalued analytic function defined by induction as