A multi-variable theta product is examined. It is shown that, under very general choices of the parameters, the quotient of two such general theta products is a root of unity. Special cases are explicitly determined. The second main theorem yields an explicit evaluation of a sum of series of cosines, which greatly generalizes one of Ramanujan’s theorems on certain sums of hyperbolic cosines.

1. For a field F and a family of central simple F algebras we prove that there exists a regular field extension E/F preserving indices of F -algebras such that all the algebras from the family are cyclic after scalar extension by E. 2. Let A be a central simple algebra over a field F of degree n with a primitive n-th root of unity ρn. We construct a quasiaffine F -variety Symb(A) such that, for...

Let p > 2 be prime, and let n,m ∈ N be given. For cyclic extensions E/F of degree p that contain a primitive pth root of unity, we show that the associated Fp[Gal(E/F )]-modules H(GE , μp) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p extension E/F , we give a more refined Fp[Gal(E/F )]-decomposition of H (GE , μp).

Vazirani and the author [Electron. J. Combin., 15 (1) (2008), R130] gave a new interpretation of what we called l-partitions, also known as (l, 0)-Carter partitions. The primary interpretation of such a partition λ is that it corresponds to a Specht module S which remains irreducible over the finite Hecke algebra Hn(q) when q is specialized to a primitive l root of unity. To accomplish this we ...

Two-dimensional displays were used to investigate the perception of object unity in 48 2-monthold and 48 4-month-old infants. The infants were habituated to a computer-generated display depicting a rod in motion behind a box. Posthabituation test trials consisted of two rod pieces (broken rod) and a complete rod, presented three times each in alternation. The 4-month-olds looked longer at the b...

We prove a Morita reduction theorem for the cyclotomic Hecke algebras Hr,p,n(q,Q) of type G(r, p, n). As a consequence, we show that computing the decomposition numbers of Hr,p,n(Q) reduces to computing the psplittable decomposition numbers (see Definition 1.1) of the cyclotomic Hecke algebras Hr′,p′,n′ (Q ), where 1 ≤ r ≤ r, 1 ≤ n ≤ n, p | p and where the parameters Q are contained in a single...

Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. A method for the construction of DSS from partitions of cyclic difference sets was introduced in [6] and applied to cyclic difference sets (n, (n−1)/2, (n−3)/4) of Paley type, where n ≡ 3 (mod 4) is a prime number. This paper develops similar constructions for prime numbers n ≡ ...

We give two new proofs of Winquist’s identity. In the first proof, we use basic properties of cube roots of unity and the Jacobi triple product identity. The latter does not use the Jacobi triple product identity.

We discuss the converse of a theorem of Potter stating that if the matrix equation AB = ωBA is satisfied with ω a primitive qth root of unity, then Aq+Bq = (A+B)q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q = 3 and a conjecture for the general case. We also present som...

Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for Fqn over Fq, where r is a prime and nk = r− 1 for some integer k. We generalize this construction by allowing arbitrary integers r with nk = φ(r), and find in many cases smaller values of k than i...