We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2) Every c.e. set of integers has a finite-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (3) All c.e. s...

For a torsion unit u of the integral group ring ZG of a finite group G, and a prime p which does not divide the order of u (but the order of G), a relation between the partial augmentations of u on the p-regular classes of G and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable G, consequences concer...

we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

If ∗ : G → G is an involution on the finite group G, then ∗ extends to an involution on the integral group ring Z[G]. In this paper, we consider whether bicyclic units u ∈ Z[G] exist with the property that the group 〈u, u∗〉, generated by u and u∗, is free on the two generators. If this occurs, we say that (u, u∗) is a free bicyclic pair. It turns out that the existence of u depends strongly upo...

Let A be a noetherian ring which is locally Macaulay. For each integer i ^ O , groups d(A) and Wi(A) are denned, each sequence of groups generalizing to higher dimensions the usual class group of an integrally closed noetherian domain. d(A) is called the ith class group of A, and Wi(A) is called the ith homologuai class group of A. The main purpose of this note is to show that both sequences of...

It is shown that any torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a trivial unit if G = P o A with P a normal Sylow p-subgroup of G and A an abelian p′-group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture ho...

We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing upper bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type...

Nys, like all polyenes, consists of two different chemical structures, namely a macrolide ring and an amino sugar. The amino sugar, a mycosamine (3-amino-3,6-dideoxy-D-mannose) is glycosidically attached to the macrolide ring. The macrolide ring consists of carbon atoms and is closed by the formation of an internal ester or a lactone. The macrolide ring is a stable rod-shaped structure with a h...

A finite group G is said to have the nilpotent decomposition property (ND) if for every element α of integral ring Z[G] one has that αe also belong Z[G], primitive central idempotent e rational algebra Q[G]. Results Hales, Passi and Wilson, Liu Passman show this fundamental in investigations multiplicative Jordan rings. If all its subgroups ND then showed SSN, is, H, Y N G, N◁H Y⊆H N⊆Y or YN no...