Anna Seigal

Local dynamics of intersections: V. I. Arnold’s theorem revisited

V. I. Arnold proved in 1993 that the intersection multiplicity between two germs of analytic subvarieties at a fixed point of a holomorphic invertible self-map remains bounded when one of the germs is dragged by iterations of the self-map. The proof is based on the Skolem-Mahler-Lech theorem on zeros in recurrent sequences.We give a different proof, based on the Noetherianity of certain algebras, which allows one to generalize Arnold’s theorem for local actions of arbitrary finitely generated commutative groups, with both discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails.

Singular vectors of orthogonally decomposable tensors

Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its singular vector tuples as a variety in a product of projective spaces.

Gram determinants of real binary tensors

A binary tensor consists of 2 n entries arranged into hypercube format 2 × 2 × ⋯ × 2. There are n ways to flatten such a tensor into a matrix of size 2 × 2 n-1. For each flattening, M, we take the determinant of its Gram matrix, det(MMT ). We consider the map that sends a tensor to its n-tuple of Gram determinants. We propose a semi-algebraic characterization of the image of this map. This offers an answer to a question raised by Hackbusch and Uschmajew concerning the higher-order singular values of tensors.

Does Antibiotic Resistance Evolve in Hospitals

Nosocomial outbreaks of bacteria are well documented. Based on these incidents, and the heavy usage of antibiotics in hospitals, it has been assumed that antibiotic resistance evolves in hospital environments. To test this assumption, we studied resistance phenotypes of bacteria collected from patient isolates at a community hospital over a 2.5-year period. A graphical model analysis shows no association between resistance and patient information other than time of arrival. This allows us to focus on time-course data. We introduce a hospital transmission model, based on negative binomial delay. Our main contribution is a statistical hypothesis test called the Nosocomial Evolution of Resistance Detector (NERD). It calculates the significance of resistance trends occurring in a hospital. It can inform hospital staff about the effects of various practices and interventions, can help detect clonal outbreaks, and is available as an R package. We applied the NERD method to each of the 16 antibiotics in the study via 16 hypothesis tests. For 13 of the antibiotics, we found that the hospital environment had no significant effect on the evolution of resistance; the hospital is merely a piece of the larger picture. The p-values obtained for the other three antibiotics (cefepime, ceftazidime, and gentamicin) indicate that particular care should be taken in hospital practices with these antibiotics. One of the three, ceftazidime, was significant after accounting for multiple hypotheses, indicating a trend of decreased resistance for this drug.