72 research outputs found
Symmetric matrices related to the Mertens function
In this paper we explore a family of congruences over from which
one builds a sequence of symmetric matrices related to the Mertens function.
From the results of numerical experiments, we formulate a conjecture about
the growth of the quadratic norm of these matrices, which implies the Riemann
hypothesis. This suggests that matrix analysis methods may come to play a more
important role in this classical and difficult problem.Comment: Version submitted to LAA; some new reference
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
In this article we continue our analysis of Schr\"odinger operators on
arbitrary graphs given as certain Laplace operators. In the present paper we
give the proof of the composition rule for the scattering matrices. This
composition rule gives the scattering matrix of a graph as a generalized star
product of the scattering matrices corresponding to its subgraphs. We perform a
detailed analysis of the generalized star product for arbitrary unitary
matrices. The relation to the theory of transfer matrices is also discussed
- …