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We propose a method for constructing 9-variable cryptographic Boolean functions from the iterates of 5-variable cellular automata rules. We then analyze, for important cryptographic properties of 5-variable cellular automata rules, how they are preserved after extension to 9-variable Boolean functions. For each cryptographic property, we analyze the proportion of 5-variable cellular automata rules that preserve it for each of the 48 affine equivalence classes. more...

We study the volume of the critical clusters for the percolation of the level sets of the Gaussian free field on metric graphs. On $\mathbb{Z}^d$ below the upper-critical dimension $d=6$, we show that the largest such cluster in a box of side length $r$ has volume of order $r^{\frac{d+2}{2}}$, as conjectured by Werner in arXiv:2002.11487. This is in contrast to the mean-field regime $d>6$, where this volume is of order $r^4$. We further obtain precise asymptotic tails for the volume of the critical cluster of the origin, and a lower bound on the tail of the volume of the near-critical cluster of the origin below the upper-critical dimension. Our proof extends to any graph with polynomial volume growth and polynomial decay of the Green's function as long as the critical one-arm probability decays as the square root of the Green's function, which is satisfied in low enough dimension., Comment: 35 pages, 1 figure more...

Deep Metric Learning (DML) loss functions traditionally aim to control the forces of separability and compactness within an embedding space so that the same class data points are pulled together and different class ones are pushed apart. Within the context of DML, a softmax operation will typically normalize distances into a probability for optimization, thus coupling all the push/pull forces together. This paper proposes a potential new class of loss functions that operate within a euclidean domain and aim to take full advantage of the coupled forces governing embedding space formation under a softmax. These forces of compactness and separability can be boosted or mitigated within controlled locations at will by using a warping function. In this work, we provide a simple example of a warping function and use it to achieve competitive, state-of-the-art results on various metric learning benchmarks., Comment: Preprint more...

Medical images are often acquired in different settings, requiring harmonization to adapt to the operating point of algorithms. Specifically, to standardize the physical spacing of imaging voxels in heterogeneous inference settings, images are typically resampled before being processed by deep learning models. However, down-sampling results in loss of information, whereas upsampling introduces redundant information leading to inefficient resource utilization. To overcome these issues, we propose to condition segmentation models on the voxel spacing using hypernetworks. Our approach allows processing images at their native resolutions or at resolutions adjusted to the hardware and time constraints at inference time. Our experiments across multiple datasets demonstrate that our approach achieves competitive performance compared to resolution-specific models, while offering greater flexibility for the end user. This also simplifies model development, deployment and maintenance. Our code is available at https://github.com/ImFusionGmbH/HyperSpace., Comment: Accepted at MICCAI 2024 more...

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. Under the sole assumption that its sign clusters do not percolate, we derive an extension of Lupu's formula for the two-point function at criticality. We then focus on the low-dimensional case $0< \nu < \frac{\alpha}{2}$, where $\alpha$ governs the polynomial volume growth of $G$ and $\nu$ the decay rate of the Green's function on $G$. In particular, this includes the benchmark case ${G}=\mathbb{Z}^3$, for which $\alpha=3$ and $\nu= \alpha-2=1$. We prove under these assumptions that the critical one-arm probability decays with distance $R$ like $R^{-\frac{\nu}{2}}$, up to multiplicative constants., Comment: 15 pages more...

We consider a class of percolation models where the local occupation variables have long-range correlations decaying as a power law $\sim r^{-a}$ at large distances $r$, for some $0< a< d$ where $d$ is the underlying spatial dimension. For several of these models, we present both, rigorous analytical results and matching simulations that determine the critical exponents characterizing the fixed point associated to their phase transition, which is of second order. The exact values we obtain are rational functions of the two parameters $a$ and $d$ alone, and do not depend on the specifics of the model., Comment: 6 pages, 4 figures more...

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu>1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu>1$ and essentially optimal when $\nu=1$. This extends previous results from arXiv:2101.05801., Comment: 21 pages more...

In this study, we investigate the combination of indicators, including performance, behavioral engagement, and emotional engagement, to identify students experiencing difficulties. We analyzed data from two primary sources: digital traces extracted from th e Learning Management System (LMS) and images captured by students' webcams. The digital traces provided insights into students' interactions with the educational content, while the images were utilized to analyze their emotional expressions during learnin g activities. By utilizing real data collected from students at a French engineering school, recorded during the 2022 2023 academic year, we observed a correlation between positive emotional states and improved academic outcomes. These preliminary findings support the notion that emotions play a crucial role in differentiating between high achieving and low achieving students., Comment: 5 pages, 2 figures more...

The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball., Comment: 68 pages, 2 figures more...

Let $X$ be a random walk on the torus of side length $N$ in dimension $d\geq 3$ with uniform starting point, and $t_{\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $\alpha > 0$, the set $\mathcal{L}^{\alpha}$ of $\alpha$-late points consists of those points not visited by $X$ at time $\alpha t_{\text{cov}}$. We prove the existence of a value $\alpha_* \in (\frac12,1)$ across which $\mathcal{L}^{\alpha}$ trivialises as follows: for all $\alpha > \alpha_*$ and $\epsilon\geq N^{-c}$ there exists a coupling of $\mathcal{L}^\alpha$ and two occupation sets $\mathcal{B}^{\alpha_\pm}$ of i.i.d. Bernoulli fields having the same density as $\mathcal{L}^{\alpha\pm \epsilon}$, which is asymptotic to $N^{-(\alpha\pm\epsilon)d}$, with the property that the inclusion $ \mathcal{B}^{\alpha_+} \subseteq \mathcal{L}^{\alpha} \subseteq \mathcal{B}^{\alpha_-}$ holds with high probability as $N \to \infty$. On the contrary, when $\alpha \leq \alpha_*$ there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at $\alpha_*$ corresponds to the (dis-)appearance of `double-points' (i.e. neighboring pairs of points) in $\mathcal{L}^\alpha$. We further describe the law of $\mathcal{L}^{\alpha}$ for $\alpha>\frac12$ by adding independent patterns to $\mathcal{B}^{\alpha_{\pm}}$. In dimensions $d \geq 4$ these are exactly all two-point sets. When $d=3$ one must also include all connected three-point sets, but no other., Comment: 68 pages, 2 figures more...

Automatic vertebra localization and identification in CT scans is important for numerous clinical applications. Much progress has been made on this topic, but it mostly targets positional localization of vertebrae, ignoring their orientation. Additionally, most methods employ heuristics in their pipeline that can be sensitive in real clinical images which tend to contain abnormalities. We introduce a simple pipeline that employs a standard prediction with a U-Net, followed by a single graph neural network to associate and classify vertebrae with full orientation. To test our method, we introduce a new vertebra dataset that also contains pedicle detections that are associated with vertebra bodies, creating a more challenging landmark prediction, association and classification task. Our method is able to accurately associate the correct body and pedicle landmarks, ignore false positives and classify vertebrae in a simple, fully trainable pipeline avoiding application-specific heuristics. We show our method outperforms traditional approaches such as Hungarian Matching and Hidden Markov Models. We also show competitive performance on the standard VerSe challenge body identification task., Comment: 13 pages, 4 figures more...

Multimodal image registration is a challenging but essential step for numerous image-guided procedures. Most registration algorithms rely on the computation of complex, frequently non-differentiable similarity metrics to deal with the appearance discrepancy of anatomical structures between imaging modalities. Recent Machine Learning based approaches are limited to specific anatomy-modality combinations and do not generalize to new settings. We propose a generic framework for creating expressive cross-modal descriptors that enable fast deformable global registration. We achieve this by approximating existing metrics with a dot-product in the feature space of a small convolutional neural network (CNN) which is inherently differentiable can be trained without registered data. Our method is several orders of magnitude faster than local patch-based metrics and can be directly applied in clinical settings by replacing the similarity measure with the proposed one. Experiments on three different datasets demonstrate that our approach generalizes well beyond the training data, yielding a broad capture range even on unseen anatomies and modality pairs, without the need for specialized retraining. We make our training code and data publicly available., Comment: This preprint was submitted to MICCAI 2023. The Version of Record of this contribution will be published in Springer LNCS more...

Generalized Stieltjes constants $\gamma$ n (a) are the coecients in the Laurent series for the Hurwitz-zeta function $\zeta$(s, a) at the pole s = 1. Many authors proved formulas for these constants. In this paper, using a recurrence between ($\zeta$(s + j, a)) j and proved by the author, we prove a general result which contains some of these formulas as particular cases. more...

In this manuscript we describe the investigation of the SARS-CoV2 membrane fusion timescale by means of small-angle neutron scattering (SANS) using hydrogen/deuterium contrast variation. After the successful production of virus-like vesicles and human-host-cell-like vesicles we were able to follow the fusion of the respective vesicles in real-time. This was done using deuterated and protonated phospholipids in the vesicles in a neutron-contrast matched solvent. The vesicles were identical apart from either the presence or absence of the SARS-CoV2 spike protein. The human-host-cell-like vesicles were carrying an ACE2 receptor protein in all cases. In case of the absence of the spike protein a fusion over several hours was observed in agreement with literature, with a time constant of 4.5 h. In comparison, there was not time-evolution, but immediate fusion of the vesicles when the spike protein was present. Those two figures, fusion over several hours and fusion below 10 s corresponding to the absence or presence of the spike protein allow an upper-limit estimate for the fusion times of virus-like vesicles with the SARS-CoV2 spike protein of 10 s. This very fast fusion, when compared to the case without spike protein it is a factor of 2500, can also help to explain why infection with SARS-CoV2 can be so effective and fast. Studying spike protein variants using our method may explain differences in transmissibility between SARS-CoV2 strains. In addition, the model developed here can potentially be applied to any enveloped virus., Comment: 15 pages, 7 figures, 2 tables more...

Deep learning image processing models have had remarkable success in recent years in generating high quality images. Particularly, the Improved Denoising Diffusion Probabilistic Models (DDPM) have shown superiority in image quality to the state-of-the-art generative models, which motivated us to investigate their capability in the generation of the synthetic electrocardiogram (ECG) signals. In this work, synthetic ECG signals are generated by the Improved DDPM and by the Wasserstein GAN with Gradient Penalty (WGAN-GP) models and then compared. To this end, we devise a pipeline to utilize DDPM in its original $2D$ form. First, the $1D$ ECG time series data are embedded into the $2D$ space, for which we employed the Gramian Angular Summation/Difference Fields (GASF/GADF) as well as Markov Transition Fields (MTF) to generate three $2D$ matrices from each ECG time series, which when put together, form a $3$-channel $2D$ datum. Then $2D$ DDPM is used to generate $2D$ $3$-channel synthetic ECG images. The $1D$ ECG signals are created by de-embedding the $2D$ generated image files back into the $1D$ space. This work focuses on unconditional models and the generation of \emph{Normal Sinus Beat} ECG signals exclusively, where the Normal Sinus Beat class from the MIT-BIH Arrhythmia dataset is used in the training phase. The \emph{quality}, \emph{distribution}, and the \emph{authenticity} of the generated ECG signals by each model are quantitatively evaluated and compared. Our results show that in the proposed pipeline and in the particular setting of this paper, the WGAN-GP model is consistently superior to DDPM in all the considered metrics. more...