A Dedekind algebra is an ordered pair (B, h) where B is a nonempty set and h is a "similarity transformation" on B. Among the Dedekind algebras is the sequence of positive integers. Each Dedekind algebra can be decomposed into a family of disjointed, countable subalgebras which are called the configurations of the algebra. There are many isomorphic types of configurations. Each Dedekind algebra...

On arbitrary topological groups a natural finitely additive measure can be defined via compactifications. It is closely related to Hartman’s concept of uniform distribution on non-compact groups (cf. [Ha]). Applications to several situations are possible. Some results of M. Paštéka and other authors on uniform distribution with respect to translation invariant finitely additive probability meas...

This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that P (gcd(z1, z2) = 1) = 1 ζQ(i)(2) where z1, z2 ∈ Z[i] are randomly chosen and...

Remark 1.6. In [Mor06], this result is stated without proof. Our goal here is to indicate that the result follows without any of the machinations appearing in §3.2 of [Mor06] (where the result is stated). Ayoub has shown that if the Krull dimension of k is ≥ 2, then the unstable A-connectivity property will fail, thus the strong A-invariance property cannot hold over 2-dimensional rings. Howeve...

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We ...

This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers, Z, to the field of r...

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale (φ,Γ)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (φ,Γ)-modules over the Robba ring, of the Euler-Poincaré characteristic formula and Tate local duality for p-adic represent...

Lecture notes for a series of talks given in Bonn, June 2015. Most of the topics covered touched in one way or another on the role of power operations in elliptic cohomology. In June of 2015 I gave a series of six lectures (the Felix Klein lectures) in Bonn. These are some of my lecture notes for those talks. I had hoped to polish them more carefully, but that hasn’t happend yet, and at this po...