Classifying Rotationally-Closed Languages Having Greedy Universal Cycles

Let T(n,k) role= presentation \u3eT(n,k) be the set of strings of length n role= presentation \u3en over the alphabet Σ={1,2,…,k} role= presentation \u3eΣ={1,2,…,k}. A universal cycle for T(n,k) role= presentation \u3eT(n,k) can be constructed using a greedy algorithm: start with the string kn role= presentation \u3ekn, and continually append the least symbol possible without repeating a substring of length n role= presentation \u3en. This construction also creates universal cycles for some subsets S⊆T(n,k) role= presentation \u3eS⊆T(n,k); we will classify all such subsets that are closed under rotations

On Onp

Generalizing John Conway\u27s construction of the Field On2, we give the minimal definitions of addition and multiplication that turn the ordinals into a Field of characteristic p, for any prime p. We then analyze the structure of the resulting Field, which we will call Onp

On prime power order elements of general linear groups

We will classify the absolutely irreducible subgroups G⩽GLd(q) role= presentation \u3e which are not realizable modulo scalars over any proper subfield of GF(q) role= presentation \u3e, which are “nearly simple”, and which contain elements of prime power order greater than d/3 role= presentation \u3e