These notes are a gloss on the first chapter of Walter Rudin’s Fourier Analysis on Groups, and may be helpful to someone reading Rudin. The results I do prove are proved in more detail than they are in Rudin. I caution that before reading the first chapter of that book it is know about the Gelfand transform on commutative Banach algebras because results from that are used without even stating t...

Let M be an o–minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n–dimensional group definable in M. We show the following: the o–minimal fundamental group of G is isomorphic to Z; for each k > 0, the k–torsion subgroup of G is isomorphic to (Z/kZ), and the o–minimal cohomology algebra over Q of G is isomorphic to the exterior algebra over Q with ...

In this paper we introduce the notion of Algebraic (Trapdoor) One Way Functions, which, roughly speaking, captures and formalizes many of the properties of number-theoretic one-way functions. Informally, a (trapdoor) one way function F : X → Y is said to be algebraic if X and Y are (finite) abelian cyclic groups, the function is homomorphic i.e. F (x) · F (y) = F (x · y), and is ringhomomorphic...

Quantum algorithms are sequences of abstract operations, performed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum alg...

Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we ...

Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p > 2h − 2 and a given tilting module has h...

It is an open problem if an elementary p-group of rank k ≥ 3 does admit full-rank normalized factorization into two of its subsets such that one of the factors has p elements. The paper provides an answer in the p ≤ 7 special case.

Given a group G acting on a set S, Mobius inversion over the lattice of subgroups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup. By taking S to be a set of subsets, partitions, or permutations congruences for binomial and multinomial coefficients. Stirling numbers of both kinds, and various other combinatorial sequences are derived. Congruenc...