Extremal classical interpolation problems (matrix case)

AbstractIn this paper, we try to find amongst the solutions w(z) of the corresponding interpolation problem the solution which satisfies an additional extremal condition. We show that degenerate interpolation problems play an important role in the theory of extremal interpolation problems. At the end of the paper we accomplish a comparison of our approach with former known approaches

Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case

Two-dimensional Hamiltonian systems

This survey article contains various aspects of the direct and inverse spectral problem for twodimensional Hamiltonian systems, that is, two dimensional canonical systems of homogeneous differential equations of the form Jy'(x) = -zH(x)y(x); x ∈ [0;L); 0 < L ≤ ∞; z ∈ C; with a real non-negative definite matrix function H ≥ 0 and a signature matrix J, and with a standard boundary condition of the form y1(0+) = 0. Additionally it is assumed that Weyl's limit point case prevails at L. In this case the spectrum of the canonical system is determined by its Titchmarsh-Weyl coefficient Q which is a Nevanlinna function, that is, a function which maps the upper complex half-plane analytically into itself. In this article an outline of the Titchmarsh-Weyl theory for Hamiltonian systems is given and the solution of the direct spectral problem is shown. Moreover, Hamiltonian systems comprehend the class of differential equations of vibrating strings with a non-homogenous mass-distribution function as considered by M.G. Krein. The inverse spectral problem for two{dimensional Hamiltonian systems was solved by L. de Branges by use of his theory of Hilbert spaces of entire functions, showing that each Nevanlinna function is the Titchmarsh-Weyl coefficient of a uniquely determined normed Hamiltonian. More detailed results of this connection for e.g. systems with a semibounded or discrete or finite spectrum are presented, and also some results concerning spectral perturbation, which allow an explicit solution of the inverse spectral problem in many cases

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

Paried Cauchy matrices

AbstractInvertibility, kernel description, and inversion of some Cauchy-type matrices (“paired Cauchy matrices”) are reduced to a rational interpolation problem. The result is applied to some problems appearing in connection with the solution of nonlinear integrable equations. Furthermore, connections between paired Cauchy matrices and paired Vandermonde and Loewner matrices are presented