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 world needs unity. It more and people to unite give full play their professional skills make things better. Innovative thinking has always been the mode advocated by designers, not result that can be achieved a deliberate mode. contains variety of modes. is an efficient unity analysis intuition, rationality sensibility. only complex, but also whole brain thinking. This reason for birth desi...

Indirect taxes on transportation activities that pollute can correct externalities and close the gaps between private and social costs. However, policy makers often find such Pigou taxes difficult to implement because of political resistance due to possibly adverse affects on equity. For this reason it is important to assess the distributional aspects of environmental levies. This article estim...

There is no single answer to the unity-multiplicity problem regarding the self for the simple reason that the term "self" is used by too many different theorists in too many different ways. In fact, there are several important substantive topic areas that need to be distinguished and studied scientifically. The topic areas I examine in this article are reflexivity, unit coherence, agency, and s...

n what follows, I shall address the question concerning the ultimate premise of the Transcendental Analytic in Kant’s Critique of Pure Reason. I believe the candidates can be reduced to two dominant but opposing principles, one grounded in immanent time-consciousness and the other in the unity of consciousness. In pursuing this goal, I intend to view consciousness through Leibnizian lenses by f...

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.