Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of G. We outline an efficient bijective proof of this theorem, by studying a canonical finite abelian group attached to G whose order is equal to the value of same matrix determinant. More specifically, we show how one can efficiently compute a ...

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe wellrounded full-rank sublattices of Z, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of suc...

Blood group determinants have been found in five carcinoembryonic antigen (CEA) preparations with a lectin and antibody-binding assay using polyethylene glycol 6000 to separate free from bound radiolabeled antigen. The assay described gives excellent sensitivity for the binding or inhibition of binding of various lectins or antibodies to CEA. One of the CEA preparations investigated has an A1 d...

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lück-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the f...

Recapitulation The origin of the representation theory of finite groups can be traced back to a correspondence between R. Dedekind and F. G. Frobenius that took place in April of 1896. The present article is based on several lectures given by the author in 1996 in commemoration of the centennial of this occasion. In Part I of this article we recounted the story of how Dedekind proposed to Frobe...