The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

Desingularization of vortices for the Euler equation

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation -\eps^2 \Delta u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.Comment: 40 page

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

In vitro evaluation of a novel bioreactor based on an integral oxygenator and a spirally wound nonwoven polyester matrix for hepatocyte culture as small aggregates

BACKGROUND/AIMS: The development of custom-made bioreactors for use as a bioartificial liver (BAL) is considered to be one of the last challenges on the road to successful temporary extracorporeal liver support therapy. We devised a novel bioreactor (patent pending) which allows individual perfusion of high density cultured hepatocytes with low diffusional gradients, thereby more closely resembling the conditions in the intact liver lobuli. METHODS: The bioreactor consists of a spirally wound nonwoven polyester matrix, i.e. a sheet-shaped, three-dimensional framework for hepatocyte immobilization and aggregation, and of integrated hydrophobic hollow-fiber membranes for decentralized oxygen supply and CO2 removal. Medium (plasma in vivo) was perfused through the extrafiber space and therefore in direct hepatocyte contact. Various parameters were assessed over a period of 4 days including galactose elimination, urea synthesis, lidocaine elimination, lactate/pyruvate ratios, amino acid metabolism, pH, the last day being reserved exclusively for determination of protein secretion. RESULTS: Microscopic examination of the hepatocytes revealed cytoarchitectural characteristics as found in vivo. The biochemical performance of the bioreactor remained stable over the investigated period. The urea synthesizing capacity of hepatocytes in the bioreactor was twice that of hepatocytes in monolayer cultures. Flow sensitive magnetic resonance imaging (MRI) revealed that the bioreactor construction ensured medium flow through all parts of the device irrespective of its size. CONCLUSIONS: The novel bioreactor showed encouraging efficiency. The device is easy to manufacture with scale-up to the liver mass required for possible short-term support of patients in hepatic failur