Counting Fiedler pencils with repetitions

We introduce a new notation based on diagrams to deal with Fiedler pencils with repetitions (FPR), and use it to solve several counting problems. In particular, we give explicit recurrences to count the number of FPRs of a given degree d, the number of symmetric, palindromic and antipalindromic ones (where the latter two structures are intended in the sense of [5]). We relate these structures to the presence of symmetries in the associated diagrams

When is a matrix unitary or Hermitian plus low rank?

Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing complexity once the matrix has been reduced, for instance, to tridiagonal or Hessenberg form. Recently, fast and reliable eigensolvers dealing with low‐rank perturbations of unitary and Hermitian matrices have been proposed. These structured eigenvalue problems appear naturally when computing roots, via confederate linearizations, of polynomials expressed in, for example, the monomial or Chebyshev basis. Often, however, it is not known beforehand whether or not a matrix can be written as the sum of a Hermitian or unitary matrix plus a low‐rank perturbation. In this paper, we give necessary and sufficient conditions characterizing the class of Hermitian or unitary plus low‐rank matrices. The number of singular values deviating from 1 determines the rank of a perturbation to bring a matrix to unitary form. A similar condition holds for Hermitian matrices; the eigenvalues of the skew‐Hermitian part differing from 0 dictate the rank of the perturbation. We prove that these relations are linked via the Cayley transform. Then, based on these conditions, we identify the closest Hermitian or unitary plus rank k matrix to a given matrix A, in Frobenius and spectral norm, and give a formula for their distance from A. Finally, we present a practical iteration to detect the low‐rank perturbation. Numerical tests prove that this straightforward algorithm is effective.status: publishe

Factoring block Fiedler Companion Matrices

When Fiedler published his “A note on Companion matrices” in 2003 in Linear Algebra and its Applications, he could not have foreseen the significance of this elegant factorization of a companion matrix into essentially two-by-two Gaus- sian transformations, which we will name (scalar) elementary Fiedler factors. Since then, researchers extended these results and studied the various resulting lineariza- tions, the stability of Fiedler companion matrices, factorizations of block companion matrices, Fiedler pencils, and even looked at extensions to non-monomial bases. In this chapter, we introduce a new way to factor block Fiedler companion matrices into the product of scalar elementary Fiedler factors. We use this theory to prove that, e.g., a block (Fiedler) companion matrix can always be written as the product of several scalar (Fiedler) companion matrices. We demonstrate that this factoriza- tion in terms of elementary Fiedler factors can be used to construct new lineariza- tions. Some linearizations have notable properties, such as low band-width, or allow for factoring the coefficient matrices into unitary-plus-low-rank matrices. Moreover, we will provide bounds on the low-rank parts of the resulting unitary-plus-low-rank decomposition. To present these results in an easy-to-understand manner we rely on the flow-graph representation for Fiedler matrices recently proposed by Del Corso and Poloni in Linear Algebra and its Applications, 2017.status: publishe

Association of maternal hypertension and chorioamnionitis with preterm outcomes

OBJECTIVES: We compared the relative effect of hypertensive disorders of pregnancy and chorioamnionitis on adverse neonatal outcomes in very preterm neonates, and studied whether gestational age (GA) modulates these effects. METHODS: A cohort of neonates 23 to 30 weeks' GA, born in 2008 to 2011 in 82 hospitals adhering to the Italian Neonatal Network, was analyzed. Infants born from mothers who had hypertensive disorders (N = 2096) were compared with those born after chorioamnionitis (N = 1510). Statistical analysis employed logistic models, adjusting for GA, hospital, and potential confounders. RESULTS: Overall mortality was higher after hypertension than after chorioamnionitis (odds ratio [OR], 1.39; 95% confidence interval [CI], 1.08-1.80), but this relationship changed across GA weeks; the OR for hypertension was highest at low GA, whereas from 28 weeks' GA onward, mortality was higher for chorioamnionitis. For other outcomes, the relative risks were constant across GA; infants born after hypertension had an increased risk for bronchopulmonary dysplasia (OR, 2.20; 95% CI, 1.68-2.88) and severe retinopathy of prematurity (OR, 1.48; 95% CI, 1.02-2.15), whereas there was a lower risk for early-onset sepsis (OR, 0.25; 95% CI, 0.19-0.34), severe intraventricular hemorrhage (OR, 0.65; 95% CI, 0.48-0.88), periventricular leukomalacia (OR, 0.70; 95% CI, 0.48-1.01), and surgical necrotizing enterocolitis or gastrointestinal perforation (OR, 0.47; 95% CI, 0.31-0.72). CONCLUSIONS: Mortality and other adverse outcomes in very preterm infants depend on antecedents of preterm birth. Hypertension and chorioamnionitis are associated with different patterns of outcomes; for mortality, the effect changes across GA weeks. Copyright \uc2\ua9 2014 by the American Academy of Pediatrics