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

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

e+e- --> nu nu-bar A in the two-Higgs-doublet model

We compute the cross section for e+e- --> nu nu-bar A, where A is the CP-odd scalar, in the general CP-conserving type-II two-Higgs-doublet model. We sum the contributions from the ``t-channel'' e+e- --> nu nu-bar W W --> nu nu-bar A graphs and ``s-channel'' e+e- --> Z A --> nu nu-bar A graphs, including their interference. Higgs-triangle graphs and all box diagrams are included. For many parameter choices, especially those in the decoupling region of parameter space (light h and m_A, m_H, m_H^+ > 2 m_Z), the Higgs-triangle and box diagrams are found to be of minor importance, the main contributing loops being the top and bottom quark triangle diagrams. The predicted cross section is rather small for tan beta > 2 and/or m_A > 2 m_t. However, we also show that if parameters are chosen corresponding to large Higgs self-couplings then the Higgs-triangle graphs can greatly enhance the cross section. We also demonstrate that the SUSY-loop corrections to the b b-bar A coupling could be such as to greatly enhance this coupling, resulting in an enhanced nu nu-bar A cross section. Complete cross section expressions are given in the Appendices.Comment: 25 pages, 13 figures, tex file requires axodraw; v2: minor changes, to appear in PR

The CP-conserving two-Higgs-doublet model: the approach to the decoupling limit

A CP-even neutral Higgs boson with Standard-Model-like couplings may be the lightest scalar of a two-Higgs-doublet model. We study the decoupling limit of the most general CP-conserving two-Higgs-doublet model, where the mass of the lightest Higgs scalar is significantly smaller than the masses of the other Higgs bosons of the model. In this case, the properties of the lightest Higgs boson are nearly indistinguishable from those of the Standard Model Higgs boson. The first non-trivial corrections to Higgs couplings in the approach to the decoupling limit are also evaluated. The importance of detecting such deviations in precision Higgs measurements at future colliders is emphasized. We also clarify the case in which a neutral Higgs boson can possess Standard-Model-like couplings in a regime where the decoupling limit does not apply. The two-Higgs-doublet sector of the minimal supersymmetric model illustrates many of the above features.Comment: 54 pages, 2 tables, revtex4 format, some new material added (including elegant forms for the three-Higgs and four-Higgs couplings) and typographical errors fixe

Whole-genome sequencing reveals host factors underlying critical COVID-19

Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2–4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genes—including reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)—in critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease