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...

the subject matter of this article is not to determine the priority of reason to heart or privacy of rationality to love, but find out the answers to these questions: what is role of reason and rational knowledge in attaining mystical intuitive visions? how far does the horizon of rational knowledge extend from perspective of ibn arabi? given the fact that mysticism is arena for esoteric vision...

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].

How are general informal reasoning skills acquired? Little research has been done on this topic. Two hypotheses dominate. According to the strong situated learning hypothesis, there are no general informal reasoning skills (only contextor domainspecific skills) and so nothing can be done to improve them. According to the practice hypothesis, general informal reasoning can be improved through in...