A partial difference set having parameters (n2, r(n − 1), n + r2 − 3r, r2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n2, r(n + 1),−n + r2 + 3r, r2 + r) is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory o...

Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). We consider algebraic properties of the resulting generating function ΦA(p, q). For q = −1, the non-zero roots in p must be roots of unity and for q > 0, real negative.

We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and inve...

For any odd prime power q, all (q&q+1)th roots of unity clearly lie in the extension field Fq6 of the Galois field Fq of q elements. It is easily shown that none of these roots of unity have trace &2, and the only such roots of trace &3 must be primitive cube roots of unity which do not belong to Fq . Here the trace is taken from Fq6 to Fq . Computer based searching verified that indeed &2 and ...

Let p be an odd prime and let F be arbitrary field of characteristic not p, containing a primitive pth root of unity ζ . In this paper, we prove a criterion, giving the obstructions to realizability of p-groups as Galois groups over F , having a factor-group of the kind H ×Cp . We apply this to the non-abelian groups of orders p3 and p4. Where it is possible, we give a description of all Galois...

Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H(SL1(D),R/Z) is a cyclic p-group whose order is bounded from below by the number of p-power roots of unity in F , unless D is a quaternion algebra over Q2. In this paper we give an ex...

We conjecture that, evaluated at the root of unity exp(2π √ −1/r) instead of the standard exp(π √ −1/r), the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. This reveals a different asymptotic behavior of the relevant quantum invariants than that of Witten’s inva...