Implicit QR with compression

AbstractIn this paper, we elaborate on the implicit shifted QR eigenvalue algorithm given in [D.A. Bini, P. Boito, Y. Eidelman, L. Gemignani, I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl. 432 (2010), 2006–2031]. The algorithm is substantially simplified and speeded up while preserving its numerical robustness. This allows us to obtain a potentially important advance towards a proof of its backward stability together with both cost reductions and implementative benefits

Algebraic construction of the Darboux matrix revisited

We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Backlund transformation, based on different lambda-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symmetric N-soliton formulas in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulas for N-soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations.Comment: Review paper (61 pages). To be published in the Special Issue "Nonlinearity and Geometry: Connections with Integrability" of J. Phys. A: Math. Theor. (2009), devoted to the subject of the Second Workshop on Nonlinearity and Geometry ("Darboux Days"), Bedlewo, Poland (April 2008

The feasibility of a multi‐professional training to improve how health care professionals deliver different news to families during pregnancy and at birth

Background: In the United Kingdom, pregnant women are offered foetal anomaly screening to assess the chance of their baby being born with eleven different conditions. How health care professionals (HCPs) deliver news about a child having a congenital anomaly affects how it is received and processed by parents. We refer to this news as different news. Methods: We conducted a mixed methods evaluation of a training intervention to improve how HCPs deliver different news. Twenty‐six HCPs self‐completed pretraining and posttraining questionnaires on skills, knowledge, and attitudes related to delivering different news. Qualitative interviews were conducted with eight HCPs. Quantitative data were analysed using descriptive statistics, the paired t test to compare the pre and post scores and estimate the difference between pre and post scores, and the 95% confidence interval. Qualitative data were analysed using framework analysis guided by the Theoretical Domains Framework (TDF). Results: The training intervention was both feasible and acceptable. HCPs indicated that it enhanced or consolidated their knowledge and skills, covered topics relevant to their practice, and that they would recommend it to colleagues. Participants particularly valued integration of the voice of parents with lived experience in the training. Significant increase in mean scores were observed in confidence to deliver different news (2.81, 95% CI [2.43, 3.19] to 4.28, 95% CI [4.09, 4.47]; p < .001) and skills to deliver different news (3.00, 95% CI [2.64, 3.36] to 4.36, 95% CI [4.13, 4.59]; p < .001). HCPs reported feeling more confident in their ability to provide sensitive, responsive, balanced care to families. Conclusions: The significant improvements in confidence and skills reported by HCPs suggest that the training may be effective in equipping HCPs to minimize the distress, anxiety, and depression associated with receiving different news. This represents a key aspect of the prevention of mental ill health across the life course