Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis Keeler) have been incorporated. The formulation of some results has been improve

Non-Extensive Bose-Einstein Condensation Model

The imperfect Boson gas supplemented with a gentle repulsive interaction is completely solved. In particular it is proved that it has non-extensive Bose-Einstein condensation, i.e., there is condensation without macroscopic occupation of the ground state (k=0) level

Routing for analog chip designs at NXP Semiconductors

During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP. This resulted in an heuristic approach, which we presented at the end of the week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

Some Global Characteristics of the Galactic Globular Cluster System

The relations between the luminosities $M_{V}$, the metallicities $[Fe/H]$, the Galactocentric radii $R$, and the central concentration indices $c$ of Galactic globular clusters are discussed. It is found that the most luminous clusters rarely have collapsed cores. The reason for this might be that the core collapse time scales for such populous clusters are greater than the age of the Galaxy. Among those clusters, for which the structure has not been modified by core collapse, there is a correlation between central concentration and integrated luminosity, in the sense that the most luminous clusters have the strongest central concentration. The outermost region of the Galaxy with $R>10$ kpc was apparently not able to form metal-rich $([Fe/H]>-1.0)$ globular clusters, whereas such clusters (of which Ter 7 is the prototype) were able to form in some nearby dwarf spheroidal galaxies. It is not yet clear how the popular hypothesis that globular clusters were initially formed with a single power law mass spectrum can be reconciled with the observation that both (1) Galactic globular clusters with $R>80$ kpc, and (2) the globulars associated with the Sagittarius dwarf, appear to have bi-modal luminosity functions.Comment: 15 pages, 1 figur

New Direct Observational Evidence for Kicks in SNe

We present an updated list of direct strong evidence in favour of kicks being imparted to newborn neutron stars. In particular we discuss the new cases of evidence resulting from recent observations of the X-ray binary Circinus X-1 and the newly discovered binary radio pulsar PSR J1141-6545. We conclude that the assumption that neutron stars receive a kick velocity at their formation is unavoidable (van den Heuvel & van Paradijs 1997).Comment: 2 pages, to appear in the proceedings of the IAU Colloq. 177 "Pulsar Astronomy - 2000 and beyond

Isospectrality and heat content

We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.Comment: 18 page