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Both structural and practical parameter non-identifiability present fundamental challenges when using mathematical models to interpret data. This issue is particularly acute in complex, applied areas such as the life sciences or engineering, where determining appropriate model complexity is challenging. While several approaches exist for diagnosing and resolving parameter non-identifiability, including symbolic methods, profile likelihood analysis, and sloppiness analysis, these approaches have distinct limitations and are rarely combined. We present an integrated approach called Invariant Image Reparameterisation (IIR) that incorporates key elements of these methods in a new way. Our approach replaces symbolic computations with numerical calculations at a single reference estimate and an invariance condition that determines when this local calculation holds globally. Parameter combinations determined by this method are naturally ordered by degree of identifiability, and this supports model reduction by replacing a practically non-identified model with a structurally non-identified approximate model. This approximate model can be further parameterised in terms of identified parameters only. By treating parameter combinations determined by our approach as interest parameters within our established likelihood-based Profile-Wise Analysis (PWA) framework, we incorporate uncertainty quantification in terms of likelihood profiles and confidence sets. We provide a Julia library on GitHub (https://github.com/omaclaren/reparam) demonstrating our methodology across a range of mathematical models., Comment: 25 pages + supplementary material

The integration of Large Language Models (LLMs) into healthcare settings has gained significant attention, particularly for question-answering tasks. Given the high-stakes nature of healthcare, it is essential to ensure that LLM-generated content is accurate and reliable to prevent adverse outcomes. However, the development of robust evaluation metrics and methodologies remains a matter of much debate. We examine the performance of publicly available LLM-based chatbots for menopause-related queries, using a mixed-methods approach to evaluate safety, consensus, objectivity, reproducibility, and explainability. Our findings highlight the promise and limitations of traditional evaluation metrics for sensitive health topics. We propose the need for customized and ethically grounded evaluation frameworks to assess LLMs to advance safe and effective use in healthcare.

Well-mixed chemical reaction networks (CRNs) contain many distinct chemical species with copy numbers that fluctuate in correlated ways. While those correlations are typically monitored via Monte Carlo sampling of stochastic trajectories, there is interest in systematically approximating the joint distribution over the exponentially large number of possible microstates using tensor networks or tensor trains. We exploit the tensor network strategy to determine when the steady state of a seven-species gene toggle switch CRN model supports bistability as a function of two decomposition rates, both parameters of the kinetic model. We highlight how the tensor network solution captures the effects of stochastic fluctuations, going beyond mean field and indeed deviating meaningfully from a mean-field analysis. The work furthermore develops and demonstrates several technical advances that will allow steady-states of broad classes of CRNs to be computed in a manner conducive to parameter exploration. We show that the steady-state distributions can be computed via the ordinary density matrix renormalization group (DMRG) algorithm despite having a non-Hermitian rate operator with a small spectral gap, we illustrate how that steady-state distribution can be efficiently projected to an order parameter that identifies bimodality, and we employ excited-state DMRG to calculate a relaxation timescale for the bistability.

Background: Healthcare has many manual processes that can benefit from automation and augmentation with Generative Artificial Intelligence (AI), the medical billing and coding process. However, current foundational Large Language Models (LLMs) perform poorly when tasked with generating accurate International Classification of Diseases, 10th edition, Clinical Modification (ICD-10-CM) and Current Procedural Terminology (CPT) codes. Additionally, there are many security and financial challenges in the application of generative AI to healthcare. We present a strategy for developing generative AI tools in healthcare, specifically for medical billing and coding, that balances accuracy, accessibility, and patient privacy. Methods: We fine tune the PHI-3 Mini and PHI-3 Medium LLMs using institutional data and compare the results against the PHI-3 base model, a PHI-3 RAG application, and GPT-4o. We use the post operative surgical report as input and the patients billing claim the associated ICD-10, CPT, and Modifier codes as the target result. Performance is measured by accuracy of code generation, proportion of invalid codes, and the fidelity of the billing claim format. Results: Both fine-tuned models performed better or as well as GPT-4o. The Phi-3 Medium fine-tuned model showed the best performance (ICD-10 Recall and Precision: 72%, 72%; CPT Recall and Precision: 77%, 79%; Modifier Recall and Precision: 63%, 64%). The Phi-3 Medium fine-tuned model only fabricated 1% of ICD-10 codes and 0.6% of CPT codes generated. Conclusions: Our study shows that a small model that is fine-tuned on domain-specific data for specific tasks using a simple set of open-source tools and minimal technological and monetary requirements performs as well as the larger contemporary consumer models., Comment: 21 pages, 3 figures, 2 tables

Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any $k \ge 2$, there exist a family of $k$ closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent., Comment: 43 pages. v2: Minor corrections

Ultracold gases of atoms from Main Group III (Group 13) of the Periodic Table, also known as "triel elements," have great potential for a new generation of quantum matter experiments. The first magneto-optical trap of a triel element (indium) was recently realized, but more progress is needed before a triel is ready for modern quantum science experiments. Cutting edge quantum science can be performed with atoms that are cooled to the 10 uK level or below, prepared in pure quantum states, and optically trapped. Here we report the achievement of all three of these milestones in atomic indium. First, we perform polarization gradient cooling of an indium gas to 15 uK. Second, we spin polarize the gas into a single hyperfine sublevel of either the $5P_{1/2}$ indium ground state or the $5P_{3/2}$ metastable state. Third, we confine indium in a 1064 nm optical lattice, achieving a 3 s trap lifetime. With these results, indium is now a candidate for a next generation quantum research platform., Comment: 8 pages, 6 figures. Includes supplemental material with 13 pages and 10 figures

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay-constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. We review the determinations of nucleon charges from the matrix elements of both isovector and flavour-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model., Comment: 435 pages, 53 Figures, 190 tables. arXiv admin note: substantial text overlap with arXiv:2111.09849, arXiv:1902.08191, some corrections and updated references

This paper introduces Moonshine, a family of speech recognition models optimized for live transcription and voice command processing. Moonshine is based on an encoder-decoder transformer architecture and employs Rotary Position Embedding (RoPE) instead of traditional absolute position embeddings. The model is trained on speech segments of various lengths, but without using zero-padding, leading to greater efficiency for the encoder during inference time. When benchmarked against OpenAI's Whisper tiny-en, Moonshine Tiny demonstrates a 5x reduction in compute requirements for transcribing a 10-second speech segment while incurring no increase in word error rates across standard evaluation datasets. These results highlight Moonshine's potential for real-time and resource-constrained applications., Comment: 7 pages, 6 figures, 3 tables

The rapid growth of virtual reality (VR) has led to increased use of social VR platforms for interaction. However, these platforms lack adequate features to support blind and low vision (BLV) users, posing significant challenges in navigation, visual interpretation, and social interaction. One promising approach to these challenges is employing human guides in VR. However, this approach faces limitations with a lack of availability of humans to serve as guides, or the inability to customize the guidance a user receives from the human guide. We introduce an AI-powered guide to address these limitations. The AI guide features six personas, each offering unique behaviors and appearances to meet diverse user needs, along with visual interpretation and navigation assistance. We aim to use this AI guide in the future to help us understand BLV users' preferences for guide forms and functionalities.

Numerical models of geothermal reservoirs typically depend on hundreds or thousands of unknown parameters, which must be estimated using sparse, noisy data. However, these models capture complex physical processes, which frequently results in long run-times and simulation failures, making the process of estimating the unknown parameters a challenging task. Conventional techniques for parameter estimation and uncertainty quantification, such as Markov chain Monte Carlo (MCMC), can require tens of thousands of simulations to provide accurate results and are therefore challenging to apply in this context. In this paper, we study the ensemble Kalman inversion (EKI) algorithm as an alternative technique for approximate parameter estimation and uncertainty quantification for geothermal reservoir models. EKI possesses several characteristics that make it well-suited to a geothermal setting; it is derivative-free, parallelisable, robust to simulation failures, and in many cases requires far fewer simulations to provide an accurate characterisation of the posterior than conventional uncertainty quantification techniques such as MCMC. We illustrate the use of EKI in a reservoir modelling context using a combination of synthetic and real-world case studies. Through these case studies, we also demonstrate how EKI can be paired with flexible parametrisation techniques capable of accurately representing prior knowledge of the characteristics of a reservoir and adhering to geological constraints, and how the algorithm can be made robust to simulation failures. Our results demonstrate that EKI provides a reliable and efficient means of obtaining accurate parameter estimates for large-scale, two-phase geothermal reservoir models, with appropriate characterisation of uncertainty.

A search for proton decay into $e^+/\mu^+$ and a $\eta$ meson has been performed using data from a 0.373 Mton$\cdot$year exposure (6050.3 live days) of Super-Kamiokande. Compared to previous searches this work introduces an improved model of the intranuclear $\eta$ interaction cross section, resulting in a factor of two reduction in uncertainties from this source and $\sim$10\% increase in signal efficiency. No significant data excess was found above the expected number of atmospheric neutrino background events resulting in no indication of proton decay into either mode. Lower limits on the proton partial lifetime of $1.4\times\mathrm{10^{34}~years}$ for $p\rightarrow e^+\eta$ and $7.3\times\mathrm{10^{33}~years}$ for $p\rightarrow \mu^+\eta$ at the 90$\%$ C.L. were set. These limits are around 1.5 times longer than our previous study and are the most stringent to date.

This study reports a comprehensive environmental scan of the generative AI (GenAI) infrastructure in the national network for clinical and translational science across 36 institutions supported by the Clinical and Translational Science Award (CTSA) Program led by the National Center for Advancing Translational Sciences (NCATS) of the National Institutes of Health (NIH) at the United States. With the rapid advancement of GenAI technologies, including large language models (LLMs), healthcare institutions face unprecedented opportunities and challenges. This research explores the current status of GenAI integration, focusing on stakeholder roles, governance structures, and ethical considerations by administering a survey among leaders of health institutions (i.e., representing academic medical centers and health systems) to assess the institutional readiness and approach towards GenAI adoption. Key findings indicate a diverse range of institutional strategies, with most organizations in the experimental phase of GenAI deployment. The study highlights significant variations in governance models, with a strong preference for centralized decision-making but notable gaps in workforce training and ethical oversight. Moreover, the results underscore the need for a more coordinated approach to GenAI governance, emphasizing collaboration among senior leaders, clinicians, information technology staff, and researchers. Our analysis also reveals concerns regarding GenAI bias, data security, and stakeholder trust, which must be addressed to ensure the ethical and effective implementation of GenAI technologies. This study offers valuable insights into the challenges and opportunities of GenAI integration in healthcare, providing a roadmap for institutions aiming to leverage GenAI for improved quality of care and operational efficiency.

We characterise the set of fundamental groups for which there exist $n$-manifolds that are $h$-cobordant (hence homotopy equivalent) but not simple homotopy equivalent, when $n$ is sufficiently large. In particular, for $n \ge 12$ even, we show that examples exist for any finitely presented group $G$ such that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands on previous work, where we constructed the first examples of even-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent. Our construction is based on doubles of thickenings, and a key ingredient of the proof is a formula for the Whitehead torsion of a homotopy equivalence between such manifolds., Comment: 25 pages. Following the suggestion of a referee, this paper has been extracted from arXiv:2312.00322

Low-light image enhancement remains a challenging task in computer vision, with existing state-of-the-art models often limited by hardware constraints and computational inefficiencies, particularly in handling high-resolution images. Recent foundation models, such as transformers and diffusion models, despite their efficacy in various domains, are limited in use on edge devices due to their computational complexity and slow inference times. We introduce ExpoMamba, a novel architecture that integrates components of the frequency state space within a modified U-Net, offering a blend of efficiency and effectiveness. This model is specifically optimized to address mixed exposure challenges, a common issue in low-light image enhancement, while ensuring computational efficiency. Our experiments demonstrate that ExpoMamba enhances low-light images up to 2-3x faster than traditional models with an inference time of 36.6 ms and achieves a PSNR improvement of approximately 15-20% over competing models, making it highly suitable for real-time image processing applications.

Optical coherence tomography (OCT) and confocal microscopy are pivotal in retinal imaging, offering distinct advantages and limitations. In vivo OCT offers rapid, non-invasive imaging but can suffer from clarity issues and motion artifacts, while ex vivo confocal microscopy, providing high-resolution, cellular-detailed color images, is invasive and raises ethical concerns. To bridge the benefits of both modalities, we propose a novel framework based on unsupervised 3D CycleGAN for translating unpaired in vivo OCT to ex vivo confocal microscopy images. This marks the first attempt to exploit the inherent 3D information of OCT and translate it into the rich, detailed color domain of confocal microscopy. We also introduce a unique dataset, OCT2Confocal, comprising mouse OCT and confocal retinal images, facilitating the development of and establishing a benchmark for cross-modal image translation research. Our model has been evaluated both quantitatively and qualitatively, achieving Fr\'echet Inception Distance (FID) scores of 0.766 and Kernel Inception Distance (KID) scores as low as 0.153, and leading subjective Mean Opinion Scores (MOS). Our model demonstrated superior image fidelity and quality with limited data over existing methods. Our approach effectively synthesizes color information from 3D confocal images, closely approximating target outcomes and suggesting enhanced potential for diagnostic and monitoring applications in ophthalmology., Comment: 30 pages, 11 figures, 5 tables

The ability to make accurate predictions with quantified uncertainty provides a crucial foundation for the successful management of a geothermal reservoir. Conventional approaches for making predictions using geothermal reservoir models involve estimating unknown model parameters using field data, then propagating the uncertainty in these estimates through to the predictive quantities of interest. However, the unknown parameters are not always of direct interest; instead, the predictions are of primary importance. Data space inversion (DSI) is an alternative methodology that allows for the efficient estimation of predictive quantities of interest, with quantified uncertainty, that avoids the need to estimate model parameters entirely. In this paper, we illustrate the applicability of DSI to geothermal reservoir modelling. We first review the processes of model calibration, prediction and uncertainty quantification from a Bayesian perspective, and introduce data space inversion as a simple, efficient technique for approximating the posterior predictive distribution. We then introduce a modification of the typical DSI algorithm that allows us to sample directly and efficiently from the DSI approximation to the posterior predictive distribution, and apply the algorithm to two model problems in geothermal reservoir modelling. We evaluate the accuracy and efficiency of our DSI algorithm relative to other common methods for uncertainty quantification and study how the number of reservoir model simulations affects the resulting approximation to the posterior predictive distribution. Our results demonstrate that data space inversion is a robust and efficient technique for making predictions with quantified uncertainty using geothermal reservoir models, providing a useful alternative to more conventional approaches.

In the realm of medical image fusion, integrating information from various modalities is crucial for improving diagnostics and treatment planning, especially in retinal health, where the important features exhibit differently in different imaging modalities. Existing deep learning-based approaches insufficiently focus on retinal image fusion, and thus fail to preserve enough anatomical structure and fine vessel details in retinal image fusion. To address this, we propose the Topology-Aware Graph Attention Network (TaGAT) for multi-modal retinal image fusion, leveraging a novel Topology-Aware Encoder (TAE) with Graph Attention Networks (GAT) to effectively enhance spatial features with retinal vasculature's graph topology across modalities. The TAE encodes the base and detail features, extracted via a Long-short Range (LSR) encoder from retinal images, into the graph extracted from the retinal vessel. Within the TAE, the GAT-based Graph Information Update (GIU) block dynamically refines and aggregates the node features to generate topology-aware graph features. The updated graph features with base and detail features are combined and decoded as a fused image. Our model outperforms state-of-the-art methods in Fluorescein Fundus Angiography (FFA) with Color Fundus (CF) and Optical Coherence Tomography (OCT) with confocal microscopy retinal image fusion. The source code can be accessed via https://github.com/xintian-99/TaGAT., Comment: 11 pages, 2 figures, accepted by MICCAI 2024

Despite recent strides made by AI in image processing, the issue of mixed exposure, pivotal in many real-world scenarios like surveillance and photography, remains inadequately addressed. Traditional image enhancement techniques and current transformer models are limited with primary focus on either overexposure or underexposure. To bridge this gap, we introduce the Unified-Exposure Guided Transformer (Unified-EGformer). Our proposed solution is built upon advanced transformer architectures, equipped with local pixel-level refinement and global refinement blocks for color correction and image-wide adjustments. We employ a guided attention mechanism to precisely identify exposure-compromised regions, ensuring its adaptability across various real-world conditions. U-EGformer, with a lightweight design featuring a memory footprint (peak memory) of only $\sim$1134 MB (0.1 Million parameters) and an inference time of 95 ms (9.61x faster than the average), is a viable choice for real-time applications such as surveillance and autonomous navigation. Additionally, our model is highly generalizable, requiring minimal fine-tuning to handle multiple tasks and datasets with a single architecture., Comment: Under submission

Competing interactions in low-dimensional materials can produce nearly degenerate electronic and structural phases. We investigate the staircase of structural phase transitions in layered IrTe$_2$ for which a number of potential transition mechanisms have been postulated. The spatial coexistence of multiple phases on the micron scale has prevented a detailed analysis of the electronic structure. By exploiting micro-ARPES obtained with synchrotron radiation we extract the electronic structure of the multiple structural phases in IrTe$_2$ in order to address the mechanism underlying the phase transitions. We find direct evidence of lowered energy states that appear in the low-temperature phases, states previously predicted by \textit{ab initio} calculations and extended here. Our results validate a proposed scenario of bonding and anti-bonding states as the driver of the phase transitions., Comment: None

Dysregulation of the tumour suppressor gene Adenomatous Polyposis Coli (APC) is a canonical step in colorectal cancer development. Curiously, most colorectal tumours carry biallelic mutations that result in only partial loss of APC function, suggesting that a "just-right" level of APC inactivation, and hence Wnt signalling, provides the optimal conditions for tumorigenesis. Mutational processes act variably across the APC gene, which could contribute to the bias against complete APC inactivation. Thus the selective consequences of partial APC loss are unclear. Here we propose a mathematical model to quantify the tumorigenic effect of biallelic APC genotypes, controlling for somatic mutational processes. Analysing sequence data from >2500 colorectal cancers, we find that APC genotypes resulting in partial protein function confer about 50 times higher probability of progressing to cancer compared to complete APC inactivation. The optimal inactivation level varies with anatomical location and additional mutations of Wnt pathway regulators. We use this context dependency to assess the regulatory effect of secondary Wnt drivers in combination with APC in vivo, and provide evidence that mutant AMER1 combines with APC genotypes that lead to relatively low Wnt. The fitness landscape of APC inactivation is consistent across microsatellite unstable and POLE-deficient colorectal cancers and tumours in patients with Familial Adenomatous Polyposis, suggesting a general "just-right" optimum, and pointing to Wnt hyperactivation as a potential cancer vulnerability., Comment: Main: 32 pages, 7 figures Supplements: 17 pages, 7 figures

This work focuses on the analysis of the mean longitudinal magnetic field as a stellar activity tracer in the context of small exoplanet detection and characterisation in radial-velocity (RV) surveys. We use SDO/HMI filtergrams to derive Sun-as-a-star magnetic field measurements, and show that the mean longitudinal magnetic field is an excellent rotational period detector and a useful tracer of the solar magnetic cycle. To put these results into context, we compare the mean longitudinal magnetic field to three common activity proxies derived from HARPS-N Sun-as-a-star data: the full-width at half-maximum, the bisector span and the S-index. The mean longitudinal magnetic field does not correlate with the RVs and therefore cannot be used as a one-to-one proxy. However, with high cadence and a long baseline, the mean longitudinal magnetic field outperforms all other considered proxies as a solar rotational period detector, and can be used to inform our understanding of the physical processes happening on the surface of the Sun. We also test the mean longitudinal magnetic field as a "stellar proxy" on a reduced solar dataset to simulate stellar-like observational sampling. With a Gaussian Process regression analysis, we confirm that the solar mean longitudinal magnetic field is the most effective of the considered indicators, and is the most efficient rotational period indicator over different levels of stellar activity. This work highlights the need for polarimetric time series observations of stars., Comment: 23 pages, 27 figures

Agent-based models (ABM) provide an excellent framework for modeling outbreaks and interventions in epidemiology by explicitly accounting for diverse individual interactions and environments. However, these models are usually stochastic and highly parametrized, requiring precise calibration for predictive performance. When considering realistic numbers of agents and properly accounting for stochasticity, this high dimensional calibration can be computationally prohibitive. This paper presents a random forest based surrogate modeling technique to accelerate the evaluation of ABMs and demonstrates its use to calibrate an epidemiological ABM named CityCOVID via Markov chain Monte Carlo (MCMC). The technique is first outlined in the context of CityCOVID's quantities of interest, namely hospitalizations and deaths, by exploring dimensionality reduction via temporal decomposition with principal component analysis (PCA) and via sensitivity analysis. The calibration problem is then presented and samples are generated to best match COVID-19 hospitalization and death numbers in Chicago from March to June in 2020. These results are compared with previous approximate Bayesian calibration (IMABC) results and their predictive performance is analyzed showing improved performance with a reduction in computation.

We report the discovery and validation of HD 21520 b, a transiting planet found with TESS and orbiting a bright G dwarf (V=9.2, $T_{eff} = 5871 \pm 62$ K, $R_{\star} = 1.04\pm 0.02\, R_{\odot}$). HD 21520 b was originally alerted as a system (TOI-4320) consisting of two planet candidates with periods of 703.6 and 46.4 days. However, our analysis supports instead a single-planet system with an orbital period of $25.1292\pm0.0001$ days and radius of $2.70 \pm 0.09\, R_{\oplus}$. Three full transits in sectors 4, 30 and 31 match this period and have transit depths and durations in agreement with each other, as does a partial transit in sector 3. We also observe transits using CHEOPS and LCOGT. SOAR and Gemini high-resolution imaging do not indicate the presence of any nearby companions, and MINERVA-Australis and CORALIE radial velocities rule out an on-target spectroscopic binary. Additionally, we use ESPRESSO radial velocities to obtain a tentative mass measurement of $7.9^{+3.2}_{-3.0}\, M_{\oplus}$, with a 3-$\sigma$ upper limit of 17.7 $M_{\oplus}$. Due to the bright nature of its host and likely significant gas envelope of the planet, HD 21520 b is a promising candidate for further mass measurements and for atmospheric characterization., Comment: Submitted to MNRAS

We obtain a partial classification of the finite groups $G$ for which the integral group ring $\mathbb{Z} G$ has projective cancellation, i.e. for which $P \oplus \mathbb{Z} G \cong Q \oplus \mathbb{Z} G$ implies $P \cong Q$ for projective $\mathbb{Z} G$-modules $P$ and $Q$. In particular, we determine when projective cancellation holds for a finite group with no exceptional binary polyhedral quotients. To do this, we prove a cancellation theorem based on a relative version of the Eichler condition. We then use a group theoretic argument to precisely determine the class of groups not covered by this result. The final classification is then obtained by applying results of Swan, Chen and Bley-Hofmann-Johnston which show failure of projective cancellation for certain groups., Comment: 39 pages. Minor update to account for an inaccuracy in the article of Bley-Hofmann-Johnston (arXiv:2407.02294) which has now been fixed. The group I x C2 = SmallGroup(240, 94) fails SFC, but was previously stated as having SFC. The statement of Theorem A has been adjusted accordingly

Transition path theory (TPT) offers a powerful formalism for extracting the rate and mechanism of rare dynamical transitions between metastable states. Most applications of TPT either focus on systems with modestly sized state spaces or use collective variables to try to tame the curse of dimensionality. Increasingly, expressive function approximators like neural networks and tensor networks have shown promise in computing the central object of TPT, the committor function, even in very high dimensional systems. That progress prompts our consideration of how one could use such a high dimensional function to extract mechanistic insight. Here, we present and illustrate a straightforward but powerful way to track how individual dynamical coordinates evolve during a reactive event. The strategy, which involves marginalizing the reactive ensemble, naturally captures the evolution of the dynamical coordinate's distribution, not just its mean reactive behavior.

A mathematical donut is a rectangle of integral side length with a smaller rectangle (called the hole of the donut), also of integral side length, strictly inside it and with sides of the rectangles parallel to each other, where the area of the larger rectangle is twice that of the smaller. Necessary and sufficient conditions are determined for when the hole of the donut can be rotated $90^{\circ}$ and a donut still exists, and a complete classification of all square or square-holed donuts is given, with the square donut classification being intimately related to Pythagorean triples., Comment: 9 pages