The Transcendence Degree over a Ring

For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of an algebra A over a ring R is given by calling elements of A algebraically dependent if they satisfy an algebraic equation over R whose trailing coefficient, with respect to some monomial ordering, is 1. The main result is that for a finitely generated algebra over a Noetherian Jacobson ring, the transcendence degree is equal to the Krull dimension

Stellar dust production and composition in the Magellanic Clouds

The dust reservoir in the interstellar medium of a galaxy is constantly being replenished by dust formed in the stellar winds of evolved stars. Due to their vicinity, nearby irregular dwarf galaxies the Magellanic Clouds provide an opportunity to obtain a global picture of the dust production in galaxies. The Small and Large Magellanic Clouds have been mapped with the Spitzer Space Telescope from 3.6 to 160 {\mu}m, and these wavelengths are especially suitable to study thermal dust emission. In addition, a large number of individual evolved stars have been targeted for 5-40 {\mu}m spectroscopy, revealing the mineralogy of these sources. Here I present an overview on the work done on determining the total dust production rate in the Large and Small Magellanic Clouds, as well as a first attempt at revealing the global composition of the freshly produced stardust.Comment: accepted for publication by Earth, Planets & Spac

On reconstructing n-point configurations from the distribution of distances or areas

One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations. In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples.Comment: 21 pages, late

Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector

Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where V =\gfq^n and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in \GL_n(\gfq). In fact, we determine \gfq[V \oplus V^*]^G for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of "doubly parametrized" series of group actions whose invariant rings are known to have a uniform description.Comment: 16 page

Flows on Simplicial Complexes

Given a graph $G$, the number of nowhere-zero \ZZ_q-flows $\phi_G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero \ZZ_q-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the polynomiality of the corresponding function $\phi_{\Delta}(q)$ for certain $q$ and certain subclasses of simplicial complexes.Comment: 10 pages, to appear in Discrete Mathematics and Theoretical Computer Science (proceedings of FPSAC'12

Lossless Representation of Graphs using Distributions

We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their sub-triangles. In the second part, we propose graph representations in terms of one-dimensional distributions (e.g., distribution of the node weights, sum of adjacent weights, etc.). For the case when the weights of the graph are real-valued vectors, we show that all graphs, except for a set of measure zero, are uniquely determined, up to isomorphism, from these distributions. The motivating application for this paper is the problem of browsing through large sets of graphs.Comment: 19 page