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We discuss the extension of the faithfulness question for the Burau representation of braid groups to the case of Artin--Tits groups. We prove that the Burau representation is not faithful in affine type $\tilde{A_3}$, and not faithful over several finite rings in type $D_4$, using an algorithmic approach based on categorical methods that generalize Bigelow's curve strategy outside of type $A$., Comment: Comments welcome!

We recover Reidemeister's theorem using smooth functions and transversality., Comment: My goal here was to give an easily readable proof for the Reidemeister theorem based on transversality techniques. I hope it can prove useful when learning or teaching knot theory, or as a gentle introduction to the use of transversality for knots and knotted surfaces. The paper will appear in the Bulletin of the Australian Mathematical Society

These are lecture notes from a lecture series given at CIRM in the Fall 2023. They give a down-to-earth introduction to Khovanov and Seidel's categorical representation of Artin-Tits groups, emphasizing the fact that it is all explicitly computable. Several prospective applications to (geometric) group theory are mentioned., Comment: Lecture notes from the CIRM conference "Current trends in representation theory, cluster algebras and geometry", November 2023. Comments welcome!

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.

A higher-order change-of-measure multilevel Monte Carlo (MLMC) method is developed for computing weak approximations of the invariant measures of SDE with drift coefficients that do not satisfy the contractivity condition. This is achieved by introducing a spring term in the pairwise coupling of the MLMC trajectories employing the order 1.5 strong It\^o--Taylor method. Through this, we can recover the contractivity property of the drift coefficient while still retaining the telescoping sum property needed for implementing the MLMC method. We show that the variance of the change-of-measure MLMC method grows linearly in time $T$ for all $T > 0$, and for all sufficiently small timestep size $h > 0$. For a given error tolerance $\epsilon > 0$, we prove that the method achieves a mean-square-error accuracy of $O(\epsilon^2)$ with a computational cost of $O(\epsilon^{-2} \big\vert \log \epsilon \big\vert^{3/2} (\log \big\vert \log \epsilon \big\vert)^{1/2})$ for uniformly Lipschitz continuous payoff functions and $O \big( \epsilon^{-2} \big\vert \log \epsilon \big\vert^{5/3 + \xi} \big)$ for discontinuous payoffs, respectively, where $\xi > 0$. We also observe an improvement in the constant associated with the computational cost of the higher-order change-of-measure MLMC method, marking an improvement over the Milstein change-of-measure method in the aforementioned seminal work by M. Giles and W. Fang. Several numerical tests were performed to verify the theoretical results and assess the robustness of the method., Comment: Initial version of the manuscript

This paper investigates the combination of intensity-based distance maps with boundary loss for point-supervised semantic segmentation. By design the boundary loss imposes a stronger penalty on the false positives the farther away from the object they occur. Hence it is intuitively inappropriate for weak supervision, where the ground truth label may be much smaller than the actual object and a certain amount of false positives (w.r.t. the weak ground truth) is actually desirable. Using intensity-aware distances instead may alleviate this drawback, allowing for a certain amount of false positives without a significant increase to the training loss. The motivation for applying the boundary loss directly under weak supervision lies in its great success for fully supervised segmentation tasks, but also in not requiring extra priors or outside information that is usually required -- in some form -- with existing weakly supervised methods in the literature. This formulation also remains potentially more attractive than existing CRF-based regularizers, due to its simplicity and computational efficiency. We perform experiments on two multi-class datasets; ACDC (heart segmentation) and POEM (whole-body abdominal organ segmentation). Preliminary results are encouraging and show that this supervision strategy has great potential. On ACDC it outperforms the CRF-loss based approach, and on POEM data it performs on par with it. The code for all our experiments is openly available.

In the past few years, in the context of fully-supervised semantic segmentation, several losses -- such as cross-entropy and dice -- have emerged as de facto standards to supervise neural networks. The Dice loss is an interesting case, as it comes from the relaxation of the popular Dice coefficient; one of the main evaluation metric in medical imaging applications. In this paper, we first study theoretically the gradient of the dice loss, showing that concretely it is a weighted negative of the ground truth, with a very small dynamic range. This enables us, in the second part of this paper, to mimic the supervision of the dice loss, through a simple element-wise multiplication of the network output with a negative of the ground truth. This rather surprising result sheds light on the practical supervision performed by the dice loss during gradient descent. This can help the practitioner to understand and interpret results while guiding researchers when designing new losses.

This article presents an identification benchmark based on data from a public swimming pool in operation. Such a system is both a complex process and easily understandable by all with regard to the stakes. Ultimately, the objective is to reduce the energy bill while maintaining the level of quality of service. This objective is general in scope and is not limited to public swimming pools. This can be done effectively through what is known as economic predictive control. This type of advanced control is based on a process model. It is the aim of this article and the considered benchmark to show that such a dynamic model can be obtained from operating data. For this, operational data is formatted and shared, and model quality indicators are proposed. On this basis, the first identification results illustrate the results obtained by a linear multivariable model on the one hand, and by a neural dynamic model on the other hand. The benchmark calls for other proposals and results from control and data scientists for comparison.

We prove full functoriality of Khovanov homology for tangled framed gl2 webs. We use this functoriality result to prove a strong positivity result for (orientable) surface skein algebras. The argument goes categorical and consists in proving that so-called linear complexes are stable under superposition., Comment: 71 pages. Preliminary version, comments welcome!

We construct a higher-order adaptive method for strong approximations of exit times of It\^o stochastic differential equations (SDE). The method employs a strong It\^o--Taylor scheme for simulating SDE paths, and adaptively decreases the step-size in the numerical integration as the solution approaches the boundary of the domain. These techniques turn out to complement each other nicely: adaptive time-stepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order schemes improve the approximation of the state of the diffusion process. We present two versions of the higher-order adaptive method. The first one uses the Milstein scheme as numerical integrator and two step-sizes for adaptive time-stepping: $h$ when far away from the boundary and $h^2$ when close to the boundary. The second method is an extension of the first one using the strong It\^o--Taylor scheme of order 1.5 as numerical integrator and three step-sizes for adaptive time-stepping. For any $\xi>0$, we prove that the strong error is bounded by $\mathcal{O}(h^{1-\xi})$ and $\mathcal{O}(h^{3/2-\xi})$ for the first and second method, respectively, and the expected computational cost for both methods is $\mathcal{O}(h^{-1} \log(h^{-1}))$. Theoretical results are supported by numerical examples, and we discuss the potential for extensions that improve the strong convergence rate even further., Comment: The computational cost results in Theorems 2.8 and 2.11 are improved through using a connection between the SDE and an absorbing-boundary Fokker--Planck equation. Correction of a few typos

We use multijet transversality techniques to give a presentation by generators and relations of the categories of framed tangled webs and foams., Comment: 88 pages, many figures

Causal emergence is the theory that macroscales can reduce the noise in causal relationships, leading to stronger causes at the macroscale. First identified using the effective information and later the integrated information in model systems, causal emergence has been analyzed in real data across the sciences since. But is it simply a quirk of these original measures? To answer this question we examined over a dozen popular measures of causation, all independently developed and widely used, and spanning different fields from philosophy to statistics to psychology to genetics. All showed cases of causal emergence. This is because, we prove, measures of causation are based on a small set of related "causal primitives." This consilience of independently-developed measures of causation shows that macroscale causation is a general fact about causal relationships, is scientifically detectable, and is not a quirk of any particular measure of causation. This finding sets the science of emergence on firmer ground, opening the door for the detection of intrinsic scales of function in complex systems, as well as assisting with scientific modeling and experimental interventions., Comment: 22 pages, 8 figures

Domain adaptation (DA) has drawn high interest for its capacity to adapt a model trained on labeled source data to perform well on unlabeled or weakly labeled target data from a different domain. Most common DA techniques require concurrent access to the input images of both the source and target domains. However, in practice, privacy concerns often impede the availability of source images in the adaptation phase. This is a very frequent DA scenario in medical imaging, where, for instance, the source and target images could come from different clinical sites. We introduce a source-free domain adaptation for image segmentation. Our formulation is based on minimizing a label-free entropy loss defined over target-domain data, which we further guide with a domain-invariant prior on the segmentation regions. Many priors can be derived from anatomical information. Here, a class ratio prior is estimated from anatomical knowledge and integrated in the form of a Kullback Leibler (KL) divergence in our overall loss function. Furthermore, we motivate our overall loss with an interesting link to maximizing the mutual information between the target images and their label predictions. We show the effectiveness of our prior aware entropy minimization in a variety of domain-adaptation scenarios, with different modalities and applications, including spine, prostate, and cardiac segmentation. Our method yields comparable results to several state of the art adaptation techniques, despite having access to much less information, as the source images are entirely absent in our adaptation phase. Our straightforward adaptation strategy uses only one network, contrary to popular adversarial techniques, which are not applicable to a source-free DA setting. Our framework can be readily used in a breadth of segmentation problems, and our code is publicly available: https://github.com/mathilde-b/SFDA, Comment: 18 pages, 8 figures, submitted to Elsevier Medical Image Analysis

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when (MLMC) is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the (MLMC) method with strong pairwise coupling that was developed and studied numerically on filtering problems in [{\it Chernov et al., Numer. Math., 147 (2021), 71-125}], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this feature., Comment: 20 pages, 12 figures