Let L/K be an abelian extension of number fields. We prove an uniform lower bound for the height in L∗ outside roots of unity. This lower bound depends only on the degree [L : K].

There exists a number of known constructions for perfect sequences over roots of unity. These include Frank sequences of length n over n roots of unity [Heimiller, 1961][Frank, 1962], Chu sequences of length n over n roots of unity for n odd and length n over 2n roots of unity for n even [Chu, 1972], Milewski sequences of length m overm roots of unity [Milewski, 1983], Liu-Fan sequences of leng...

We construct an invariant JM of integral homology spheres M with values in a completion Ẑ[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten-ReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhin-Turaev invariants of M . As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) a...

A k-element subset D of a finite multiplicative group G of order v is called a (v, k, λ, μ)partial difference set in G (PDS) provided that the multiset of ‘differences’ {d1d −1 2 | d1, d2 ∈ D, d1 6= d2} contains each nonidentity element of D exactly λ times and each nonidentity element in G\D exactly μ times. See [11] for background on partial difference sets. We will limit our attention to abe...

Are companies, churches, and states genuine agents? Or are they just collections of individual agents that give a misleading impression of unity? This question is important, since the answer dictates how we should explain the behaviour of these entities and whether we should treat them as responsible and accountable on the model of individual agents. Group Agency offers a new approach to that q...

A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type B analogues.

The difference between EVALP(C,D) and EVAL(C,D) is that EVALP(C,D) fixes the value of a vertex w by i. We want to prove EVALP(C,D) ≡ EVAL(C,D). It is easy to see that EVAL(C,D) ≤ EVALP(C,D). Thus, we only need to prove the other direction. First, we define the notion of a discrete unitary matrix. Definition 1. Let F ∈ C be a matrix. We say F is M-discrete unitary for some positive integer M if ...

Jesus is often described in the Gospels as a preacher. What has become his most famous sermon is recorded in Matthew 5-7 and Luke 6:17-49. The former is usually referred to as the Sermon on the Mount (note 5:1), the latter as the Sermon on the Plain (note 6:17 KJV). The following study will be concerned primarily with the unity and structure of the Matthean sermon, but a necessary preliminary i...