The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes’ type multiple FrobeniusEuler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes’ type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define gene...

The classification of the fragments of Halpern and Shoham’s logic with respect to decidability/undecidability of the satisfiability problem is now very close to the end. We settle one of the few remaining questions concerning the fragment AĀBB̄, which comprises Allen’s interval relations “meets” and “begins” and their symmetric versions. We already proved that AĀBB̄ is decidable over the class of...

We show that the presentation of affine T-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine G-varieties of complexi...

Unlike the uniform completion, the Dedekind completion of a vector lattice is not functorial. In order to repair the lack of functoriality of Dedekind completions, we enrich the signature of vector lattices with a proximity relation, thus arriving at the category pdv of proximity Dedekind vector lattices. We prove that the Dedekind completion induces a functor from the category bav of bounded a...

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P ∗) if and only if R is an artinian s...

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...

Invertibility of multiplication modules All rings are commutative with 1 and all modules are unital. Let R be a ring and M an R-module. M is called multiplication if for each submodule N of M, N=IM for some ideal I of R. Multiplication modules have recently received considerable attention during the last twenty years. In this talk we give the de nition of invertible submodules as a natural gene...

A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models of Dedekind domains called diophantine approximation groups. The arithmetic of diophantine approximation groups is defined in a way which extends the ideal-...