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This work studies collections of Hilbert space operators which possess a strict monoid structure under composition. These collections can be thought of as discrete unital semigroups for which no subset of the collection is closed under composition (apart from the trivial subset containing only the identity). We call these collections $\textit{forward operator monoids}$. Although not traditionally painted in this light, monoids of this flavor appear in many areas of analysis; as we shall see, they are intimately linked to several well-known problems. The main aim of this work is to study three classes of vectors associated to a given operator monoid: cyclic vectors, generalized inner vectors, and vectors which are both cyclic and inner (we refer to this last class as aleph vectors). We show, when they exist, aleph vectors are unique up to a multiplicative constant. We also show, under the lens of a least-squares problem, how an aleph vector can be used to characterize all cyclic and inner vectors. As an application of this theory, we consider monoids related to the Periodic Dilation Completeness Problem and the Riemann Hypothesis. After recovering some known results, we conclude by giving a further reformulation of the Riemann Hypothesis based on works of B\'{a}ez-Duarte and Noor., Comment: 18 pages, comments welcome (please send to cfelder@iu.edu)

We study the behavior of the superperiod map near the boundary of the moduli space of stable supercurves and prove that it is similar to the behavior of periods of classical curves. We consider two applications to the geometry of this moduli space in genus $2$, denoted as $\bar{\mathcal S}_2$. First, we characterize the canonical projection of $\bar{\mathcal S}_2$ in terms of its behavior near the boundary, proving in particular that $\bar{\mathcal S}_2$ is not projected. Secondly, we combine the information on superperiods with the explicit calculation of genus $2$ Mumford isomorphism, due to Witten, to study the expansion of the superstring measure for genus $2$ near the boundary. We also present the proof, due to Deligne, of regularity of the superstring measure on $\bar{\mathcal S}_g$ for any genus., Comment: 66 pages

In pulmonary tracheal segmentation, the scarcity of annotated data is a prevalent issue in medical segmentation. Additionally, Deep Learning (DL) methods face challenges: the opacity of 'black box' models and the need for performance enhancement. Our Human-Computer Interaction (HCI) based models (RS_UNet, LC_UNet, UUNet, and WD_UNet) address these challenges by combining diverse query strategies with various DL models. We train four HCI models and repeat these steps: (1) Query Strategy: The HCI models select samples that provide the most additional representative information when labeled in each iteration and identify unlabeled samples with the greatest predictive disparity using Wasserstein Distance, Least Confidence, Entropy Sampling, and Random Sampling. (2) Central line correction: Selected samples are used for expert correction of system-generated tracheal central lines in each training round. (3) Update training dataset: Experts update the training dataset after each DL model's training epoch, enhancing the trustworthiness and performance of the models. (4) Model training: The HCI model is trained using the updated dataset and an enhanced UNet version. Experimental results confirm the effectiveness of these HCI-based approaches, showing that WD-UNet, LC-UNet, UUNet, and RS-UNet achieve comparable or superior performance to state-of-the-art DL models. Notably, WD-UNet achieves this with only 15%-35% of the training data, reducing physician annotation time by 65%-85%.

We discuss the zero sets of two-variable polynomials as they relate to an approximation problem in the Hardy space on the bidisk., Comment: 21 pages; comments welcome

We study the harmonic locus consisting of the monodromy-free Schr\"odinger operators with rational potential and quadratic growth at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via the Wronskian map for Hermite polynomials. We show that the harmonic locus can also be identified with the subset of the Calogero--Moser space introduced by Wilson, which is fixed by the symplectic action of $\mathbb C^\times.$ As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing the partition in terms of the spectrum of the corresponding Moser matrix. We also compute the characters of the $\mathbb C^\times$-action at the fixed points, proving, in particular, a conjecture of Conti and Masoero. In the Appendix written by N. Nekrasov there is an alternative proof of this result, based on the space of instantons and ADHM construction., Comment: Extended version with an Appendix written by N. Nekrasov containing an alternative proof of Conti-Masoero conjecture

Each medical segmentation task should be considered with a specific AI algorithm based on its scenario so that the most accurate prediction model can be obtained. The most popular algorithms in medical segmentation, 3D U-Net and its variants, can directly implement the task of lung trachea segmentation, but its failure to consider the special tree-like structure of the trachea suggests that there is much room for improvement in its segmentation accuracy. Therefore, a research gap exists because a great amount of state-of-the-art DL algorithms are vanilla 3D U-Net structures, which do not introduce the various performance-enhancing modules that come with special natural image modality in lung airway segmentation. In this paper, we proposed two different network structures Branch-Level U-Net (B-UNet) and Branch-Level CE-UNet (B-CE-UNet) which are based on U-Net structure and compared the prediction results with the same dataset. Specially, both of the two networks add branch loss and central line loss to learn the feature of fine branch endings of the airways. Uncertainty estimation algorithms are also included to attain confident predictions and thereby, increase the overall trustworthiness of our whole model. In addition, predictions of the lung trachea based on the maximum connectivity rate were calculated and extracted during post-processing for segmentation refinement and pruning.

We study isomorphism classes of untwisted irregular singular meromorphic connections on principal bundles over (wild) Riemann surfaces, for any complex reductive structure group $G$ and polar divisor. In particular we compute the stabilisers of suitable marked points on their principal part orbits, showing the stabilisers are connected and controlled by the corresponding filtration of (Levi factors of) nested parabolic subgroups of $G$; this uniquely determines the orbits as complex homogeneous manifolds for groups of jets of principal $G$-bundle automorphisms. Moreover, when the residue is semisimple we stratify the space of orbits by the stabilisers, relating this to local wild mapping class groups and generalising the Levi stratification of a Cartan subalgebra $\mathfrak{t} \subseteq \mathfrak{g} = \operatorname{Lie}(G)$: the dense stratum corresponds to the generic setting of irregular isomonodromic deformations \`a la Jimbo-Miwa-Ueno. Then we adapt a result of Alekseev-Lachowska to deformation-quantise nongeneric orbits: the $\ast$-product involves affine-Lie-algebra modules, extending the generalised Verma modules (in the case of regular singularities) and the `singularity' modules of F.-R. (in the case of generic irregular singularities). As in the generic case, the modules contain Whittaker vectors for the Gaiotto-Teschner Virasoro pairs from irregular Liouville conformal field theory; but they now provide all the quotients which are obtained when the corresponding parameters leave the aforementioned dense strata. We also construct Shapovalov forms for the corresponding representations of truncated (holomorphic) current Lie algebras, leading to a conjectural irreducibility criterion. Finally, we use these representations to construct new flat vector bundles of vacua/covacua \`a la Wess--Zumino-Novikov-Witten, equipped with connections \`a la Knizhnik-Zamolodchikov., Comment: 112 pages, 3 figures. v2: added references and simplified bits of the exposition. Comments welcome!

Airway-related quantitative imaging biomarkers are crucial for examination, diagnosis, and prognosis in pulmonary diseases. However, the manual delineation of airway trees remains prohibitively time-consuming. While significant efforts have been made towards enhancing airway modelling, current public-available datasets concentrate on lung diseases with moderate morphological variations. The intricate honeycombing patterns present in the lung tissues of fibrotic lung disease patients exacerbate the challenges, often leading to various prediction errors. To address this issue, the 'Airway-Informed Quantitative CT Imaging Biomarker for Fibrotic Lung Disease 2023' (AIIB23) competition was organized in conjunction with the official 2023 International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI). The airway structures were meticulously annotated by three experienced radiologists. Competitors were encouraged to develop automatic airway segmentation models with high robustness and generalization abilities, followed by exploring the most correlated QIB of mortality prediction. A training set of 120 high-resolution computerised tomography (HRCT) scans were publicly released with expert annotations and mortality status. The online validation set incorporated 52 HRCT scans from patients with fibrotic lung disease and the offline test set included 140 cases from fibrosis and COVID-19 patients. The results have shown that the capacity of extracting airway trees from patients with fibrotic lung disease could be enhanced by introducing voxel-wise weighted general union loss and continuity loss. In addition to the competitive image biomarkers for prognosis, a strong airway-derived biomarker (Hazard ratio>1.5, p<0.0001) was revealed for survival prognostication compared with existing clinical measurements, clinician assessment and AI-based biomarkers., Comment: 19 pages

Given the urgency to reduce fossil fuel energy production to make climate tipping points less likely, we call for resource-aware knowledge gain in the research areas on Universe and Matter with emphasis on the digital transformation. A portfolio of measures is described in detail and then summarized according to the timescales required for their implementation. The measures will both contribute to sustainable research and accelerate scientific progress through increased awareness of resource usage. This work is based on a three-days workshop on sustainability in digital transformation held in May 2023., Comment: 20 pages, 2 figures, publication following workshop 'Sustainability in the Digital Transformation of Basic Research on Universe & Matter', 30 May to 2 June 2023, Meinerzhagen, Germany, https://indico.desy.de/event/37480

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1 < p < \infty$). For fixed $f\in H^p$ and $n\in\mathbb{N}$, the OPA of degree $n$ associated to $f$ is the polynomial which minimizes the quantity $\|qf-1\|_p$ over all complex polynomials $q$ of degree less than or equal to $n$. We begin with some examples which illustrate, when $p\neq2$, how the Banach space geometry makes these problems interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions of certain fixed point problems. Finally, using duality arguments, we provide several bounds concerning the error incurred in the OPA approximation., Comment: 28 pages

We determine the coefficient of proportionality between two multidimensional hypergeometric integrals. One of them is a solution of the dynamical difference equations associated with a Young diagram and the other is the vertex integral associated with the Young diagram. The coefficient of proportionality is the inverse of the product of weighted hooks of the Young diagram. It turns out that this problem is closely related to the question of describing the action of the center of the universal enveloping algebra of $\mathfrak{gl}_n$ on the space of Whittaker vectors in the tensor product of dual Verma modules with fundamental modules, for which we give an explicit basis of simultaneous eigenvectors., Comment: Latex, 15 pages

Matrix configurations define noncommutative spaces endowed with extra structure including a generalized Laplace operator, and hence a metric structure. Made dynamical via matrix models, they describe rich physical systems including noncommutative gauge theory and emergent gravity. Refining the construction in [1], we construct a semi-classical limit through an immersed submanifold of complex projective space based on quasi-coherent states. We observe the phenomenon of oxidation, where the resulting semi-classical space acquires spurious extra dimensions. We propose to remove this artifact by passing to a leaf of a carefully chosen foliation, which allows to extract the geometrical content of the noncommutative spaces. This is demonstrated numerically via multiple examples.

Synthetic images generated from deep generative models have the potential to address data scarcity and data privacy issues. The selection of synthesis models is mostly based on image quality measurements, and most researchers favor synthetic images that produce realistic images, i.e., images with good fidelity scores, such as low Fr\'echet Inception Distance (FID) and high Peak Signal-To-Noise Ratio (PSNR). However, the quality of synthetic images is not limited to fidelity, and a wide spectrum of metrics should be evaluated to comprehensively measure the quality of synthetic images. In addition, quality metrics are not truthful predictors of the utility of synthetic images, and the relations between these evaluation metrics are not yet clear. In this work, we have established a comprehensive set of evaluators for synthetic images, including fidelity, variety, privacy, and utility. By analyzing more than 100k chest X-ray images and their synthetic copies, we have demonstrated that there is an inevitable trade-off between synthetic image fidelity, variety, and privacy. In addition, we have empirically demonstrated that the utility score does not require images with both high fidelity and high variety. For intra- and cross-task data augmentation, mode-collapsed images and low-fidelity images can still demonstrate high utility. Finally, our experiments have also showed that it is possible to produce images with both high utility and privacy, which can provide a strong rationale for the use of deep generative models in privacy-preserving applications. Our study can shore up comprehensive guidance for the evaluation of synthetic images and elicit further developments for utility-aware deep generative models in medical image synthesis., Comment: 10 pages, 4 figures, MICCAI Early Acceptance

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1 < p < \infty$). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if $f \in H^p$ is an inner function, or if $p>2$ is an even integer, then the roots of the nontrivial OPA for $1/f$ are bounded from the origin by a distance depending only on $p$. For $p\neq 2$, these results are made possible by the novel use of a family of inequalities which are derived from a Banach space analogue of the Pythagorean theorem.

Training medical AI algorithms requires large volumes of accurately labeled datasets, which are difficult to obtain in the real world. Synthetic images generated from deep generative models can help alleviate the data scarcity problem, but their effectiveness relies on their fidelity to real-world images. Typically, researchers select synthesis models based on image quality measurements, prioritizing synthetic images that appear realistic. However, our empirical analysis shows that high-fidelity and visually appealing synthetic images are not necessarily superior. In fact, we present a case where low-fidelity synthetic images outperformed their high-fidelity counterparts in downstream tasks. Our findings highlight the importance of comprehensive analysis before incorporating synthetic data into real-world applications. We hope our results will raise awareness among the research community of the value of low-fidelity synthetic images in medical AI algorithm training., Comment: CBMS 2023

Matrix configurations coming from matrix models comprise many important aspects of modern physics. They represent special quantum spaces and are thus strongly related to noncommutative geometry. In order to establish a semiclassical limit that allows to extract their intuitive geometrical content, this thesis analyzes and refines an approach that associates a classical geometry to a given matrix configuration, based on quasi-coherent states. While, so far, the approach is only well understood for very specific cases, in this work it is reviewed and implemented on a computer, allowing the numerical investigation of deformations of these cases. It is proven that the classical space can be made into a smooth manifold immersed into complex projective space. Further, the necessity for the consideration of foliations thereof is shown in order to deal with the observed and subsequently described phenomenon called oxidation. The developed numerical methods allow the visualization of the semiclassical limit as well as quantitative calculations. Explicit examples suggest the stability under perturbations of the refined approach and highlight the physical interpretation of the construction. All this supports a better understanding of the geometrical content of arbitrary matrix configurations as well as their classical interpretation and establishes the determination of important quantities., Comment: Masters thesis, 112 pages, 46 figures

Organic neuromorphic device networks can accelerate neural network algorithms and directly integrate with microfluidic systems or living tissues. Proposed devices based on the bio-compatible conductive polymer PEDOT:PSS have shown high switching speeds and low energy demand. However, as electrochemical systems, they are prone to self-discharge through parasitic electrochemical reactions. Therefore, the network's synapses forget their trained conductance states over time. This work integrates single-device high-resolution charge transport models to simulate neuromorphic device networks and analyze the impact of self-discharge on network performance. Simulation of a single-layer nine-pixel image classification network reveals no significant impact of self-discharge on training efficiency. And, even though the network's weights drift significantly during self-discharge, its predictions remain 100\% accurate for over ten hours. On the other hand, a multi-layer network for the approximation of the circle function is shown to degrade significantly over twenty minutes with a final mean-squared-error loss of 0.4. We propose to counter the effect by periodically reminding the network based on a map between a synapse's current state, the time since the last reminder, and the weight drift. We show that this method with a map obtained through validated simulations can reduce the effective loss to below 0.1 even with worst-case assumptions. Finally, while the training of this network is affected by self-discharge, a good classification is still obtained. Electrochemical organic neuromorphic devices have not been integrated into larger device networks. This work predicts their behavior under nonideal conditions, mitigates the worst-case effects of parasitic self-discharge, and opens the path toward implementing fast and efficient neural networks on organic neuromorphic hardware.

This work explores several aspects of interpolating sequences for $\ell^p_A$, the space of analytic functions on the unit disk with $p$-summable Maclaurin coefficients. Much of this work is communicated through a Carlesonian lens. We investigate various analogues of Gramian matrices, for which we show boundedness conditions are necessary and sufficient for interpolation, including a characterization of universal interpolating sequences in terms of Riesz systems. We also discuss weak separation, giving a characterization of such sequences using a generalization of the pseudohyperbolic metric. Lastly, we consider Carleson measures and embeddings., Comment: 36 pages, 2 figures