In 1977 Kervaire and Murthy presented three conjectures regarding K0ZCpn , where Cpn is the cyclic group of order p n and p is a semi-regular prime that is p does not divide h (regular p does not divide the class number h = hh). The Mayer-Vietoris exact sequence provides the following short exact sequence 0 → Vn → PicZCpn → ClQ(ζn−1)× PicZCpn−1 → 0 Here ζn−1 is a primitive p -th root of unity. ...

The algebra su 2 is derived from two commuting quon algebras for which the parameter q is a root of unity. This leads to a polar decomposition of the shift operators J + and J − of the group SU 2 (with J + = J † − = HUr where H is Hermitean and Ur unitary). The Wigner-Racah algebra of SU 2 is developed in a new basis arising from the simultanenous diagonalization of the commuting operators J 2 ...

In his work on Wiener-Hopf determinants, A. Böttcher came across what he termed a “mysterious” identity that evaluates a certain sum of a rational function of a primitive root of unity in terms of the Barnes Gfunction, which was later generalized by Basor and Forrester. We give a direct proof of Böttcher’s identity and its many generalizations by using elements of the theory of symmetric functi...

One trivial zero phenomenon for p-adic analytic function is considered. We then prove that the first derivative of this function is essentially the Kummer class associated with p. 1. Introduction. In this paper we always fix an odd prime p > 2. For n ≥ 1, fix a p n th primitive root of unity ζ p n such that ζ

In this note, we calculate all the basic invariants of the number field K = Q(3 √ 5, ω), where ω = (−1 + √ −3)/2 is a primitive cube root of unity. Here is the notation for the fields and Galois groups to be used. Let

The quintuple identity has a long history and, as Berndt [5] points out, it is difficult to assign priority to it. It seems that a proof of the identity was first published in H. A. Schwartz’s book in 1893 [19]. Watson gave a proof in 1929 in his work on the RogersRamanujan continued fractions [20]. Since then, various proofs have appeared. To name a few, Carlitz and Subbarao gave a simple proo...

Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using Schilling’s bijection between ribbon tableaux and ribbon rigged configurations.

We must mention that the assumption that we are dealing with a Cartan matrix of finite type and a root of unity appears only at the very end (see Chapter 4). We need these assumptions in order to compare our representations with the conventional definition of the category C. All previous results are valid in more general assumptions. In particular a Cartan matrix could be arbitrary and a deform...

unity is one of the most important social issues in the holy religion of islam. unity makes possible entanglement from dependencies, success in settling social issues, and firmness in confronting conspiracies and invasions. since islam is a religion of kindness and brotherhood, it invites its followers to engage in being united, which will, however, never be achieved without resolving conflicts...