Abstract. We show that for any prime number l > 2 the minus class group of the field of the l-th roots of unity Qp(ζl) admits a finite free resolution of length 1 as a module over the ring Ẑ[G]/(1 + ι). Here ι denotes complex conjugation in G = Gal(Qp(ζl)/Qp) ∼= (Z/lZ)∗. Moreover, for the primes l ≤ 509 we show that the minus class group is cyclic as a module over this ring. For these primes we...

Explicit formulae for τ ′ r(L(p, q)) and τ(L(p, q)) are obtained for all L(p, q). There are three systems of invariants of Witten-type for closed oriented 3manifolds: 1. {τr(M), r ≥ 2; τ ′ r(M), r odd ≥ 3}, where τr was defined by Reshetikhin and Turaev [1], and τ ′ r was defined by Kirby-Melvin [2] 2. {Θr(M,A), r ≥ 1,whereA is a 2r-th primitive root of unity} defined by Blanchet, Habegger, Mas...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

On the 15th of September 2008, a power-sharing agreement was brokered between the major political parties in Zimbabwe, namely, the Tsvangirai-led Movement for Democratic Change (MDC-T), the Mutambara-led Movement for Democratic Change (MDC-M) and the then ruling Zimbabwe African National Union–Patriotic Front (ZANU-PF). This Global Political Agreement (GPA) paved the way for the establishment o...

The three Divine Abrahamic faiths i.e. Judaism, Christianity and Islam, revealed to the prophets of God, are divine religions that shape outlook of life, people’s perception, culture and civilization and enhance the relationship between God-man-universe. In this process, people who adhere to a particular religion tend to explore the phenomena through the lens of religion. However, throughout th...

In this paper, the cycles of Tanner’s (3, 5) quasi-cyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4, 6, 8, and 10 in Tanner’s (3, 5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15-th root of unity of the prime field Fp. By checking the existence of solutio...

We show that the height of a nonzero algebraic number α that lies in an abelian extension of the rationals and is not a root of unity must satisfy h(α) > 0.155097.

We construct a canonical compactification SQ g,K of the moduli of abelian varieties over Z[ζN , 1/N ] where ζN is a primitive Nth root of unity. It is very similar to, but slightly different from the compactification SQg,K in [N99]. Any degenerate abelian scheme on the boundary of SQ g,K is one of the (torically) stable quasi-abelian schemes introduced in [AN99], which is reduced and singular. ...

We give three constructions of three-class association schemes as fusion schemes of the cyclotomic scheme, two of which are primitive.