A cluster for the octagonal square-rhombus tiling is presented, which has the property that among all tilings completely covered by the cluster the perfectly quasiperiodic and eightfold symmetric ones have the highest cluster density. Since on these eightfold symmetric tilings there is considerable overlap of clusters, it seems likely that these tiling have the highest cluster density even amon...

We establish sharp bounds for the second moment of symmetric-square $L$-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where is a sum over $t_j$ in short interval. At central point $s=1/2$ $L$-function, our interval smaller than previous known results. More specifically, $|t_j|$ size $T$, $T^{1/5}$, while best was $T^{1/3}$ from work Lam. A little higher up on...

We define a semi-magic square to be a square matrix whose entries are nonnegative integers and whose rows and columns (that is, lines) sum up to the same number. A magic square is a semimagic square whose main diagonals also add up to the line sum. A symmetric magic square is a magic square which is a symmetric matrix. A pandiagonal magic square is a semi-magic square whose diagonals parallel t...

Keywords: Self-orthogonal Latin square Strongly symmetric Magic square Yang Hui type a b s t r a c t In this paper, a strongly symmetric self-orthogonal diagonal Latin square of order n with a special property (* SSSODLS(n)) is introduced. It is proved that a * SSSODLS(n) exists if and only if n ≡ 0 (mod 4) and n ̸ = 4. As an application, it is shown that there exists a Yang Hui type magic squar...