d The action of the derivation e = q ~ on the q-expansions of modular forms in characteristic p is one of the fundamental tools in the Serre/Swinnerton-Dyer theory of mod p modular forms. In this note, we extend the basic results about this action, already known for P > 5 and level one, to arbitrary p and arbitrary prime-to-p level. !. Review of modular forms in characteristic p We fix an algeb...

In [1], we proposed a model for quantized discrete general relativity with a Euclidean signature. The model was constructed by combining the structure of a certain tensor category and of a full subcategory of it with the combinatorics of a triangulated 4-manifold. The category was the representations of Uqso(4), for q a 4n-th root of unity and the subcategory the representations which are calle...

Let Q(ζ) be the cyclotomic field obtained from Q by adjoining a primitive seventh root of unity ζ. Normalized primary elements of this field are characterized and related to Jacobi sums and to solutions of a system of quadratic Diophantine equations of Dickson type involving a rational prime p ≡ 1 (mod 7). These objects and their connection are then used to give another formulation of the compl...

Let p ≡ 1 (mod 4) be a prime number and let ζ = e be a primitive root of unity. Then there exists a unique biquadratic extension eld Q(y)/Q that is a sub eld of Q(ζ). The aim of this work is to construct an algorithm for nding such y explicitly. Finally we state a general conjecture about the y we found.

In this paper, we show that the problem of computing the complex roots of unity is not as simple as it seems at rst. In particular, the formulas given in a standard programmer's reference book Knuth, Seminumerical Algorithms, 1981] are shown to be numerically unstable , giving unacceptably large error for moderate sized sequences. We give alternative formulas, which we show to be superior both ...

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].

This study was performed to evaluate the efficacy of injection of essential oil of Eugenia caryophyllata in the kid horn buds, as a new chemical technique for disbudding. Five-day-old healthy goat kids from both sexes (n = 16) were divided randomly into 4 equal groups. In groups 1, 2 and 3, 0.2 mL of clove essence and in group 4 (control) 0.2 mL of normal saline was injected into the l...