On Shapiroʼs compactness criterion for composition operators

AbstractWe give an elementary and direct proof of the identity:lim sup|w|→1−Nψ(w)1−|w|=lim sup|a|→1−(1−|a|2)‖1/(1−a¯ψ)‖H22, for any analytic self-map ψ of {z:|z|<1}; where Nψ denotes the Nevanlinna counting function of ψ. We further show that one can find analytic self-maps ψ of {z:|z|<1}, where the composition operator Cψ on the Hardy space H2 is compact, such that ‖ψn‖H2 tends to zero at an arbitrarily slow rate, as n→∞; even in the case that ψ is univalent. Among these are new examples, where Cψ is compact on H2, but not in any of the Schatten classes

Closed-Range Composition Operators on A2 and the Bloch Space

For any analytic self-map φ of {z : |z| \u3c 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cφ to be closed-range on the Bloch space B . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cφ is closed-range on the Bergman space A2 , then it is closed-range on B , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem

Closed-Range Composition Operators on A2 and the Bloch Space

For any analytic self-map φ of {z : |z| \u3c 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cφ to be closed-range on the Bloch space B . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cφ is closed-range on the Bergman space A2 , then it is closed-range on B , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem

Mathematical Musings of a Urologist

The derivatives with respect to the variable $r$ of $\pi r^2$ and $\frac{4}{3}\pi r^3$ are $2\pi r$ and $4\pi r^2$, respectively. This relates, through the derivative, the area enclosed in a circle to the length of that circle and, likewise, the volume of a sphere to the surface area of that sphere. The reasons why this works are basic to a first course in calculus. In this brief article, we expand on these ideas to shapes other than circles and spheres. Our approach is with the first year calculus student in mind

Auditory compensation for head rotation is incomplete

Hearing is confronted by a similar problem to vision when the observer moves. The image motion that is created remains ambiguous until the observer knows the velocity of eye and/or head. One way the visual system solves this problem is to use motor commands, proprioception and vestibular information. These ‘extra-retinal signals’ compensate for self movement, converting image motion into head-centred coordinates, though not always perfectly. We investigated whether the auditory system also transforms coordinates by examining the degree of compensation for head rotation when judging a moving sound. Real-time recordings of head motion were used to change the ‘movement gain’ relating head movement to source movement across a loudspeaker array. We then determined psychophysically the gain that corresponded to a perceptually-stationary source. Experiment 1 showed that the gain was small and positive for a wide range of trained head speeds. Hence listeners perceived a stationary source as moving slightly opposite to the head rotation, in much the same way that observers see stationary visual objects move against a smooth pursuit eye movement. Experiment 2 showed the degree of compensation remained the same for sounds presented at different azimuths, although the precision of performance declined when the sound was eccentric. We discuss two possible explanations for incomplete compensation, one based on differences in the accuracy of signals encoding image motion and self-movement, and one concerning statistical optimisation that sacrifices accuracy for precision. We then consider the degree to which such explanations can be applied to auditory motion perception in moving listeners

Longitudinal associations between hearing loss and general cognitive ability:The Lothian Birth Cohort 1936

Hearing impairment is associated with poorer cognitive function in later life. We tested for the potential contribution of childhood cognitive ability to this relationship. Childhood cognitive ability is strongly related to cognitive function in older age, and may be related to auditory function through its association with hearing impairment risk factors. Using data from the Lothian Birth Cohort 1936, we tested whether childhood cognitive ability predicted later-life hearing ability then whether this association was mediated by demographic or health differences. We found that childhood cognitive ability was negatively associated with hearing impairment risk at age 76 (odds ratio = .834, p = .042). However, this association was non-significant following subsequent adjustment for potentially mediating demographic and health factors. Next, we tested whether associations observed in older age between hearing impairment and general cognitive ability level or change were accounted for by childhood cognitive ability. At age 76, in the minimally adjusted model, hearing impairment was associated with poorer general cognitive ability level (β = -.119, p = .030) but was not related to decline in general cognitive ability. The former association became non-significant following additional adjustment for childhood cognitive ability (β = -.068; p = .426) suggesting that childhood cognitive ability contributes (potentially via demographic and health differences) to the association between levels of hearing and cognitive function in older age. Further work is needed to test whether early-life cognitive ability also contributes to the association (documented in previous studies) between older-age hearing impairment and cognitive decline

Acoustic, psychophysical, and neuroimaging measurements of the effectiveness of active cancellation during auditory functional magnetic resonance imaging

Functional magnetic resonance imaging (fMRI) is one of the principal neuroimaging techniques for studying human audition, but it generates an intense background sound which hinders listening performance and confounds measures of the auditory response. This paper reports the perceptual effects of an active noise control (ANC) system that operates in the electromagnetically hostile and physically compact neuroimaging environment to provide significant noise reduction, without interfering with image quality. Cancellation was first evaluated at 600 Hz, corresponding to the dominant peak in the power spectrum of the background sound and at which cancellation is maximally effective. Microphone measurements at the ear demonstrated 35 dB of acoustic attenuation [from 93 to 58 dB sound pressure level (SPL)], while masked detection thresholds improved by 20 dB (from 74 to 54 dB SPL). Considerable perceptual benefits were also obtained across other frequencies, including those corresponding to dips in the spectrum of the background sound. Cancellation also improved the statistical detection of sound-related cortical activation, especially for sounds presented at low intensities. These results confirm that ANC offers substantial benefits for fMRI research

Research to support the British Library's work on emerging formats

A report for the British Library examining the issues around the legal deposit and preservation of apps, interactive narratives and databases