Noncommutative Residues and a Characterisation of the Noncommutative Integral

We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent), i.e. as a generalised quantum limit. Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra.Comment: 15 page

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension L/K such that X has ordinary reduction at every non-archimedean place of L outside a density zero set of places.Comment: 7 page

Long Circuits and Large Euler Subgraphs

An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we study the parameterized complexity of the following Euler subgraph problems: - Large Euler Subgraph: For a given graph G and integer parameter k, does G contain an induced Eulerian subgraph with at least k vertices? - Long Circuit: For a given graph G and integer parameter k, does G contain an Eulerian subgraph with at least k edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. We find this a bit surprising because the problem of finding an induced Eulerian subgraph with exactly k vertices is known to be W[1]-hard. The complexity of the problem changes drastically on directed graphs. On directed graphs we obtained the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is solvable in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs

GRB spectral parameter modeling

Fireball model of the gamma-ray bursts (GRBs) predicts generation of numerous internal shocks, which efficiently accelerate charged particles and generate relatively small-scale stochastic magnetic and electric fields. The accelerated particles diffuse in space due to interaction with the random waves and so emit so called Diffusive Synchrotron Radiation (DSR) in contrast to standard synchrotron radiation they would produce in a large-scale regular magnetic fields. In this contribution we present key results of detailed modeling of the GRB spectral parameters, which demonstrate that the non-perturbative DSR emission mechanism in a strong random magnetic field is consistent with observed distributions of the Band parameters and also with cross-correlations between them.Comment: 3 pages; IAU symposium # 274 "Advances in Plasma Astrophysics