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This book shows how operator theory interacts with function theory in one and several variables. The authors develop the theory in detail, leading the reader to the cutting edge of contemporary research. It starts with a treatment of the theory of bounded holomorphic functions on the unit disc. Model theory and the network realization formula are used to solve Nevanlinna-Pick interpolation problems, and the same techniques are shown to work on the bidisc, the symmetrized bidisc, and other domains. The techniques are powerful enough to prove the Julia-Carathéodory theorem on the bidisc, Lempert's theorem on invariant metrics in convex domains, the Oka extension theorem, and to generalize Loewner's matrix monotonicity results to several variables. In Part II, the book gives an introduction to non-commutative function theory, and shows how model theory and the network realization formula can be used to understand functions of non-commuting matrices.
Previous studies using resting-state functional neuroimaging have revealed alterations in whole-brain images, connectome-wide functional connectivity and graph-based metrics in groups of patients with schizophrenia relative to groups of healthy controls. However, it is unclear which of these measures best captures the neural correlates of this disorder at the level of the individual patient.
Here we investigated the relative diagnostic value of these measures. A total of 295 patients with schizophrenia and 452 healthy controls were investigated using resting-state functional Magnetic Resonance Imaging at five research centres. Connectome-wide functional networks were constructed by thresholding correlation matrices of 90 brain regions, and their topological properties were analyzed using graph theory-based methods. Single-subject classification was performed using three machine learning (ML) approaches associated with varying degrees of complexity and abstraction, namely logistic regression, support vector machine and deep learning technology.
Connectome-wide functional connectivity allowed single-subject classification of patients and controls with higher accuracy (average: 81%) than both whole-brain images (average: 53%) and graph-based metrics (average: 69%). Classification based on connectome-wide functional connectivity was driven by a distributed bilateral network including the thalamus and temporal regions.
These results were replicated across the three employed ML approaches. Connectome-wide functional connectivity permits differentiation of patients with schizophrenia from healthy controls at single-subject level with greater accuracy; this pattern of results is consistent with the ‘dysconnectivity hypothesis’ of schizophrenia, which states that the neural basis of the disorder is best understood in terms of system-level functional connectivity alterations.
Constitutional scholars emphasize the importance of an enduring, stable constitutional order, which North and Weingast (1989) argue is consistent with credible commitments to sustainable fiscal policies. However, this view is controversial and has received little empirical study. We use 19th-century US state-level data to estimate relationships between constitutional design and the likelihood of a government default. Results indicate that more entrenched and less specific constitutions are associated with a lower likelihood of default.
To review the literature regarding screening for vestibular schwannoma in the context of demographic changes leading to increasing numbers of elderly patients presenting with asymmetric auditory symptoms.
A systematic review of the literature was performed, with narrative synthesis and statistical analysis of data where appropriate.
Vestibular schwannomas diagnosed in patients aged over 70 years exhibit slower growth patterns and tend to be of smaller size compared to those tumours in younger age groups. This fact, combined with reduced life expectancy, renders the probability of these tumours in the elderly requiring active treatment with surgery or stereotactic radiotherapy to be extremely low. Vestibular schwannomas in the elderly are much more likely to be managed by serial monitoring with magnetic resonance imaging. The weighted yield of magnetic resonance imaging in the diagnosis of vestibular schwannoma in all age groups is 1.18 per cent, with almost 85 scans required to diagnose 1 tumour.
An evidence-based approach to the investigation of asymmetric hearing loss and tinnitus in the elderly patient can be used to formulate guidelines for the rational use of magnetic resonance imaging in this population.
The YBJ equation (Young & Ben Jelloul, J. Marine Res., vol. 55, 1997, pp. 735–766) provides a phase-averaged description of the propagation of near-inertial waves (NIWs) through a geostrophic flow. YBJ is obtained via an asymptotic expansion based on the limit
is the Burger number of the NIWs. Here we develop an improved version, the YBJ+ equation. In common with an earlier improvement proposed by Thomas, Smith & Bühler (J. Fluid Mech., vol. 817, 2017, pp. 406–438), YBJ+ has a dispersion relation that is second-order accurate in
. (YBJ is first-order accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ+ is a Padé approximant to the exact dispersion relation and with
of order unity this is significantly more accurate than the power-series approximation of Thomas et al. (2017). Moreover, in the limit of high horizontal wavenumber
, the wave frequency of YBJ+ asymptotes to twice the inertial frequency
. This enables solution of YBJ+ with explicit time-stepping schemes using a time step determined by stable integration of oscillations with frequency
. Other phase-averaged equations have dispersion relations with frequency increasing as
(Thomas et al. 2017): in these cases stable integration with an explicit scheme becomes impractical with increasing horizontal resolution. The YBJ+ equation is tested by comparing its numerical solutions with those of the Boussinesq and YBJ equations. In virtually all cases, YBJ+ is more accurate than YBJ. The error, however, does not go rapidly to zero as the Burger number characterizing the initial condition is reduced: advection and refraction by geostrophic eddies reduces in the initial length scale of NIWs so that
increases with time. This increase, if unchecked, would destroy the approximation. We show, however, that dispersion limits the damage by confining most of the wave energy to low
. In other words, advection and refraction by geostrophic flows does not result in a strong transfer of initially near-inertial energy out of the near-inertial frequency band.