Orthogonality constraints and entropy in the SO(5)-Theory of HighT_c-Superconductivity

S.C. Zhang has put forward the idea that high-temperature-superconductors can be described in the framework of an SO(5)-symmetric theory in which the three components of the antiferromagnetic order-parameter and the two components of the two-particle condensate form a five-component order-parameter with SO(5) symmetry. Interactions small in comparison to this strong interaction introduce anisotropies into the SO(5)-space and determine whether it is favorable for the system to be superconducting or antiferromagnetic. Here the view is expressed that Zhang's derivation of the effective interaction V_{eff} based on his Hamiltonian H_a is not correct. However, the orthogonality constraints introduced several pages after this 'derivation' give the key to an effective interaction very similar to that given by Zhang. It is shown that the orthogonality constraints are not rigorous constraints, but they maximize the entropy at finite temperature. If the interaction drives the ground-state to the largest possible eigenvalues of the operators under consideration (antiferromagnetic ordering, superconducting condensate, etc.), then the orthogonality constraints are obeyed by the ground-state, too.Comment: 10 pages, no figure

Note on a Conjecture of Wegner

The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner's conjecture.Comment: 5 pages; Contributions to Algebra and Geometry, Vol.52 No1 April 201

Subgraph covers -- An information theoretic approach to motif analysis in networks

Many real world networks contain a statistically surprising number of certain subgraphs, called network motifs. In the prevalent approach to motif analysis, network motifs are detected by comparing subgraph frequencies in the original network with a statistical null model. In this paper we propose an alternative approach to motif analysis where network motifs are defined to be connectivity patterns that occur in a subgraph cover that represents the network using minimal total information. A subgraph cover is defined to be a set of subgraphs such that every edge of the graph is contained in at least one of the subgraphs in the cover. Some recently introduced random graph models that can incorporate significant densities of motifs have natural formulations in terms of subgraph covers and the presented approach can be used to match networks with such models. To prove the practical value of our approach we also present a heuristic for the resulting NP-hard optimization problem and give results for several real world networks.Comment: 10 pages, 7 tables, 1 Figur

Horizontal factorizations of certain Hasse--Weil zeta functions - a remark on a paper by Taniyama

In one of his papers, using arguments about l-adic representations, Taniyama expresses the zeta function of an abelian variety over a number field as an infinite product of modified Artin L-functions. The latter can be further decomposed as products of modified Dedekind zeta functions. After recalling Taniyama's work, we give a simple geometric proof of the resulting product formula for abelian and more general group schemes

Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions

Non-linear sigma models that arise from the supersymmetric approach to disordered electron systems contain a non-compact bosonic sector. We study the model with target space H^2, the two-hyperboloid with isometry group SU(1,1), and prove that in three dimensions moments of the fields are finite in the thermodynamic limit. Thus the non-compact symmetry SU(1,1) is spontaneously broken. The bound on moments is compatible with the presence of extended states.Comment: 21 pages, dedicated to F.J. Dyson on the occasion of his 80th birthda