Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $\phi\not\in\mathsf{Aut}(\mathbb D)$, find a non-constant inner function $\Psi$ satisfying the functional equation $\Psi\circ\phi=\tau\Psi$, where $\tau$ is a unimodular constant. We prove that this problem has a solution if and only if $\phi$ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $\tau=1$. If in addition, $\phi$ is parabolic, we prove that the problem has a solution $\Psi$ for $any$ unimodular $\tau$. Finally, we show that if $\phi$ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $\tau$ satisfy $f\circ\phi=\tau f$.

Let $\mathcal A$ be a unital operator algebra and let $\theta: \mathcal A \to B(\mathcal H)$ be a continuous unital homomorphism. We prove that for all $n \in \mathbb{N}$ and all $\beta$ in the dual space of $\mathcal{A}$, \begin{equation*} \|\theta^{(n)}\| \le \max(1, \|\theta^{(n)} + \beta^{(n)} I \|). \end{equation*} Here, $\theta^{(n)}: M_n(\mathcal A) \to B(\mathcal H^n)$ denotes the $n$-th matrix ampliation of $\theta$. This extends a result of Clou\^atre, Ostermann and Ransford (the case $n=1$), who were motivated by Crouzeix's conjecture. Our result allows us to control the completely bounded norm of $\theta$, which in turn has dilation theoretic consequences. As an application, we obtain a new proof of the similarity theorem of Okubo and Ando, which says that if $T \in B(\mathcal H)$ is an operator of class $C_\rho$, where $\rho \ge 1$, then there exists an invertible operator $S$ such that $\|S^{-1} T S \| \le 1$ and such that $\|S\| \|S^{-1}\| \le \rho$. The conclusion of the Okubo--Ando theorem implies that the inequality \[ \|f(T)\| \le \rho \|f\|_{\overline{\mathbb D}} \] holds for all polynomials $f$. A direct proof of this inequality was recently given by Clou\^atre, Ostermann and Ransford. Our main result shows that this inequality in fact holds for matrix-valued polynomials, so that the existence of the similarity $S$ follows from Paulsen's similarity theorem, which says that an operator $T$ is completely polynomially bounded with constant $\rho$ if and only if there exists an invertible operator $S$ such that $\|S^{-1} T S \| \le 1$ and such that $\|S\| \|S^{-1}\| \le \rho$. The key to proving our main result is to adapt a variational argument of Caldwell, Greenbaum and Li to matrix-valued holomorphic functions. Rather than working with biholomorphic automorphisms of the disc, we work with Potapov--M\"obius transforms $m_A$, where $A$ belongs to the unit ball of $M_n(\mathbb C)$. A slightly simplified version of our key lemma, which is shown using this technique, then reads as follows: If $R \in M_n(B(\mathcal H))$ satisfies $\|R\| > 1$ and $\|m_{A \otimes I}(R)\| \le \|R\|$ for all $A$ close to $0$ and if $x \in \mathcal H^n$ with $\|x\| = 1$ and $\|R x\| = \|R\|$, then \[ \langle R x, (A \otimes I) x \rangle = 0 \] for all $A \in M_n(\mathcal C)$.

We characterize all algebraic subsets of the tridisk that are Caratheodory sets, that is the intrinsic Caratheodory metric on the set equals the Caratheodory metric for the tridisk. We show that such sets are either retracts, or are isomorphic to one particular exceptional set.

The workshop brought together researchers in two fields, non-commutative function theory and free probability. There was a mini-course in each of these areas, and speakers in all the talks were encouraged to give expository talks that illuminated the broader reaches of their fields.

We consider pairs of anti-commuting 2p-by-2p Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of 2-by-2 blocks on which the first matrix has eigenvalues +/- x(j) and the second has eigenvalues +/- y(j). We call (x(j),y(j)) the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling.

We study the algebra A generated by the Hardy operator H and the operator Mx of multiplication by x on L-2[0,1]. We call A the Hardy-Weyl algebra. We show that its quotient by the compact operators is isomorphic to the algebra of functions that are continuous on Lambda and analytic on the interior of Lambda for a planar set Lambda = [-1,0]boolean OR D(1,1), which we call the lollipop. We find a Toeplitz-like short exact sequence for the C & lowast;-algebra generated by A. We study the operator Z=H-M-x, show that its point spectrum is (-1,0]boolean OR D(1,1), and that the eigenvalues grow in multiplicity as the points move to 0 from the left.

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function f f in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type f = φ g f=\varphi g , where g g is cyclic, φ \varphi is a contractive multiplier, and ‖ f ‖ = ‖ g ‖ \|f\|=\|g\| . In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.

The role of cerebrovascular damage in Alzheimer’s Disease (AD) has resulted in inconsistent findings particularly when analyzing longitudinal data. We introduce WMH shape analysis (WSA) that quantifies white matter hyperintensity (WMH) expansion as the distance from lesion voxels to a region of interest boundary. We sought to examine the extent and spatial topography of WMH expansion with progression to AD. Fluid-Attenuated Inversion Recovery (FLAIR) images were extracted across two timepoints from ADNI. 275 longitudinal participants were classified into groups based on amyloid and cognitive status: 99 controls (CN), 42 amyloid positive cognitively stable preclinical (PC) AD, and 134 amyloid positive cognitively impaired AD individuals. WSA was performed on 5 WMH clusters (juxtacortical, anterior corona radiata, periventricular, parietal, and occipital/posterior horn). The intensity value for each cluster ROI boundary voxels reflects the Euclidean Distance from a voxel to the closest WMH voxel. The difference in distances (DD) for each timepoint provides a metric of WMH expansion. A voxel-wise omnibus ANOVA was performed across all three groups followed by subsequent T-tests. Additionally, we evaluated the relationship between the changes in DD maps with PET amyloid summary values, hypertension, and APOE e4 status. WMH expansion was found to be significant across three groups (Fig. 1) and was primarily within two WMH clusters (p<0.05 corrected). T-tests revealed that the AD group had significantly greater expansion in the juxtacortical cluster compared to CN but not the PC group (p<0.05 corrected). However, for the posterior horn that difference was driven by the PC with greater WMH expansion compared to either the AD or CN group (p<0.05 corrected). A significant association was observed with amyloid in the juxtacortical cluster but not with hypertension. A strong trend (p = 0.06 corrected) was observed for APOE e4 status in the juxtacortical cluster associating with more copies of the e4 allele. This new approach to quantifying and analyzing WMH longitudinal expansion revealed different patterns of expansion with increased AD severity. These results implicate that WMH growth associated with AD progression is spatially specific and grows in distinct ways throughout the disease process.

AbstractWe establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

We prove that the invariant subspaces of the Hardy operator on $L^2[0,1]$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.

Considering three viral transmission routes— fomites, droplets, and aerosols— two routes have been the focus of debate about the relative role of droplets and aerosols in SARS-CoV-2 infection. We seek to quantify infection risk in an enclosed space via short-range and long-range airborne transmission to inform public health decision making. Data from five published studies were analyzed to predict relative exposure at distances of 1 m and farther, mediated by droplet size divided into two bins: ≧ 8 μm (medium and large droplets that we call “droplets”) and < 8 μm (small droplets that we call “aerosols”). The results at 1 m from an infectious individual were treated as a boundary condition to model infection risk at shorter and longer distance. At all distances, infection risk was treated as the sum of exposure to aerosols and droplets. It was assumed that number of virions is proportional to particle volume. The largest infection risk occurred close to the infectious individual, and out to approximately 1m, droplets and aerosols both contributed. Farther away, the largest risk was due to aerosols. For one model, droplet exposure disappeared at 1.8 m. Policy concerning physical distancing for meaningful infection reduction relies on exposure as a function of distance, yet within this construct particle size determines respiratory deposition. This two-fold distance effect can be used to evaluate measures such as plexiglass barriers, masking, and ventilation.

OBJECTIVE:Neurodegenerative conditions often manifest radiologically with the appearance of premature aging. Multiple sclerosis (MS) biomarkers related to lesion burden are well developed, but measures of neurodegeneration are less well-developed. The appearance of premature aging quantified by machine learning applied to structural MRI assesses neurodegenerative pathology. We assess the explanatory and predictive power of "brain age" analysis on disability in MS using a large, real-world dataset.METHODS:Brain age analysis is predicated on the over-estimation of predicted brain age in patients with more advanced pathology. We compared the performance of three brain age algorithms in a large, longitudinal dataset (>13,000 imaging sessions from >6,000 individual MS patients). Effects of MS, MS disease course, disability, lesion burden, and DMT efficacy were assessed using linear mixed effects models.RESULTS:MS was associated with advanced predicted brain age cross-sectionally and accelerated brain aging longitudinally in all techniques. While MS disease course (relapsing vs. progressive) did contribute to advanced brain age, disability was the primary correlate of advanced brain age. We found that advanced brain age at study enrollment predicted more disability accumulation longitudinally. Lastly, a more youthful appearing brain (predicted brain age less than actual age) was associated with decreased disability.INTERPRETATION:Brain age is a technically tractable and clinically relevant biomarker of disease pathology that correlates with and predicts increasing disability in MS. Advanced brain age predicts future disability accumulation.

We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random variable that takes values in the set of commuting $d$-tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the Hermitian case, we characterize the eigenvalue distribution as $n$ tends to infinity. In the non-Hermitian case, we get a formula that holds if the set is irreducible. We show that there are qualitative differences between the single matrix case and the several commuting matrices case.

We define a monomial space to be a subspace of $\ltwo$ that can be approximated by spaces that are spanned by monomial functions. We describe the structure of monomial spaces.

We characterize the common range of the adjoints of cyclic multiplication operators on the Drury--Arveson space. We show that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. To achieve this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the strictly smaller Smirnov class of the Drury-Arveson space, and that this in turn equals the common range of the adjoints of cyclic multiplication operators.

INTRODUCTION:Vascular damage in Alzheimer's disease (AD) has shown conflicting findings particularly when analyzing longitudinal data. We introduce white matter hyperintensity (WMH) longitudinal morphometric analysis (WLMA) that quantifies WMH expansion as the distance from lesion voxels to a region of interest boundary. METHODS:WMH segmentation maps were derived from 270 longitudinal fluid-attenuated inversion recovery (FLAIR) ADNI images. WLMA was performed on five data-driven WMH patterns with distinct spatial distributions. Amyloid accumulation was evaluated with WMH expansion across the five WMH patterns. RESULTS:The preclinical group had significantly greater expansion in the posterior ventricular WM compared to controls. Amyloid significantly associated with frontal WMH expansion primarily within AD individuals. WLMA outperformed WMH volume changes for classifying AD from controls primarily in periventricular and posterior WMH. DISCUSSION:These data support the concept that localized WMH expansion continues to proliferate with amyloid accumulation throughout the entirety of the disease in distinct spatial locations.

We prove an asymptotic version of the Muntz-Szasz theorem, and use it to prove that all monomial operators are vanishing preserving.(c) 2022 Elsevier GmbH. All rights reserved.

Prior studies of aging and Alzheimer disease have evaluated resting state functional connectivity (FC) using either seed-based correlation (SBC) or independent component analysis (ICA), with a focus on particular functional systems. SBC and ICA both are insensitive to differences in signal amplitude. At the same time, accumulating evidence indicates that the amplitude of spontaneous BOLD signal fluctuations is physiologically meaningful. We systematically compared covariance-based FC, which is sensitive to amplitude, vs. correlation-based FC, which is not, in affected individuals and controls drawn from two cohorts of participants including autosomal dominant Alzheimer disease (ADAD), late onset Alzheimer disease (LOAD), and age-matched controls. Functional connectivity was computed over 222 regions of interest and group differences were evaluated in terms of components projected onto a space of lower dimension. Our principal observations are: (1) Aging is associated with global loss of resting state fMRI signal amplitude that is approximately uniform across resting state networks. (2) Thus, covariance FC measures decrease with age whereas correlation FC is relatively preserved in healthy aging. (3) In contrast, symptomatic ADAD and LOAD both lead to loss of spontaneous activity amplitude as well as severely degraded correlation structure. These results demonstrate a double dissociation between age vs. Alzheimer disease and the amplitude vs. correlation structure of resting state BOLD signals. Modeling results suggest that the AD-associated loss of correlation structure is attributable to a relative increase in the fraction of locally restricted as opposed to widely shared variance.

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

Objectives Considering three viral transmission routes: fomite contact, aerial transmission by droplets, and aerial transmission by aerosols, the aerial routes have been the focus of debate about the relative role of droplets and aerosols in SARS-CoV-2 infection. We seek to quantify infection risk in an enclosed space via short-range airborne transmission from droplets and long-range risk from aerosols toward focusing public health measures.Methods Data from three published studies were analyzed to predict relative exposure at distances of 1 m and farther, mediated by droplet size divided into two bins: larger than 8 µm and smaller than 75 µm (medium droplets) and smaller than 8 µm (small droplets or aerosols). The results at 1 m from an infectious individual were treated as a boundary condition to model infection risk at greater distance. At all distances, infection risk was treated as the sum of exposure to small and medium droplets. It was assumed that number of virions is proportional to droplet volume.Results The largest infection risk (as exposure to droplet volume) came from medium droplets, close to the infectious individual out to approximately 1 m. Farther away, the largest risk was due to aerosols. For one model, medium droplet exposure disappeared at 1.8 m.Conclusions Policy concerning social distancing for meaningful infection reduction relies on droplet exposure as a function of distance, yet within this construct droplet size determines respiratory deposition. This two-fold distance effect can be used to evaluate additional measures such as plexiglass barriers and masking.### Competing Interest StatementThe authors have declared no competing interest.### Funding StatementNational Science Foundation Grant DMS 2054199 Riksbankens Jubileumsfond (Swedish Foundation for the Social Sciences and Humanities) program on Science and Proven Experience National Institutes of Health Grant R01 AG052550-01A1### Author DeclarationsI confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained.YesI confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals.YesI understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance).YesI have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable.YesAll data produced in the present work are contained in the manuscript