We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. then use these to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups quadratic twists elliptic curves.

Let \(2<n<\omega\). Then \({\sf CA}_n\) denotes the class of cylindric algebras dimension \(n\), RCA}_n\) representable \(\sf CA_n\)s, CRCA}_n\) completely CA}_n\)s, and Nr}_n{\sf CA}_{\omega}(\subseteq {\sf CA}_n\)) \(n\)-neat reducts CA}_{\omega}\)s. The elementary closure CRCA}_n\)s (\(\mathbf{K_n}\)) non-elementary At}({\sf CA}_{\omega})\) are characterized using two-player zero-sum g...

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn in the plane is at least Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the vertices and a region R of D with the fo...