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Using coarse-grained molecular dynamics simulations and a united-monomer model of PE, single well-aligned multi-lamella PE crystals grown in previous work [ACS Macro Letters 12, 808 (2023)] are deformed uniaxially to mimic tensile testing. During deformation, the crystallinity, tie-chain segments, entanglements and folds are monitored and correlated with the stress-strain behaviour and mechanical properties. At small strains, the single well-aligned PE crystals reveal a larger Young modulus when the deformation direction is perpendicular to the global stem direction. At large strains, the memory of the initial topology plays little role in the mechanical response and is observed to be completely destroyed. Short chain branching appears to suppress disentanglement and shear induced alignment of the chains during deformation. As a result, tie-chains and entanglements persist in branched systems and the peak stress at failure is found to be proportional to the change in number of tie-chains from the beginning of the brittle break to its end. Our findings suggest the remarkable mechanical properties of bimodal branched PE result directly from tie-chains, with entanglements playing a secondary role in the mechanical response.

Polyethylene (PE) chains, with CH2 groups replaced by CBr2 at regular intervals ("precision PE"), have been observed to exhibit competing polymorphs driven by a preference for quantized fold lengths by Tasaki et al. Motivated by this recent discovery, the crystallisation behaviour of such precision PE chains, 400 carbons long with CBr2 groups placed regularly at every 21st carbon, is investigated using molecular dynamics simulations. The united-monomer model of PE is extended to include dibromo groups, with steric clashes at the bromines reflected in a triple-well bending potential, demonstrating its function as a preferred fold site. Different crystallisation protocols, continuous-cooling and self-seeding, reveal remarkably different crystals. Using self-seeding, the crystalline lamellar thickness increases monotonically with temperature, in quantized multiples of the distance between dibromo units. Polymer chains are observed to fold preferentially at the dibromo groups and such groups appear to be tolerated within the crystal lamellae. On quenching the bromos assemble to form registered layers, not unlike Smectic phases observed in liquid crystals, which confirms the experimental observation of competing Form I and Form I' polymorphs.

The effect of short chain grafting on the liquid crystalline (LC) ordering of nano-crystals is investigated using molecular dynamics simulations of a coarse-grained grafted nano-rod model. Monodisperse nano-rods, with aspect ratios typical of cellulose nano-crystals (CNCs) are grafted randomly with oligomers at different grafting densities. LC ordering depends non-trivially on grafting density as the effective nano-rod shape and softness is modified. Ungrafted rods exhibit Nematic and tilted Smectic-C phases. At 25\% grafting, the addition of a few side chains entirely suppresses Smectic order and instead a persistent Nematic phase is favoured. Intermediary grafting, around 50\%, results in a Nematic and the reappearance of tilted Smectic-F phases. Heavier grafting facilitates direct transitions to either Smectic-I with extreme tilt (75\%) or an un-tilted Smectic-B (100\%). Such behaviour falls outside of current hard or soft-rod descriptions of phase-transitions in rod-like LC systems and points to undiscovered LC behaviour in both shape-purified grafted/un-grafted CNCs., Comment: 17 pages, 11 figures

In this work, we address the challenge of reconstructing the complete 12-lead ECG signal from incomplete parts of it. We focus on two main scenarii: (i) reconstructing missing signal segments within an ECG lead and (ii) recovering missing leads from a single-lead. We propose a model with a U-Net architecture trained on a novel objective function to address the reconstruction problem. This function incorporates both spatial and temporal aspects of the ECG by combining the distance in amplitude between the reconstructed and real signals with the signal trend. Through comprehensive assessments using both a real-life dataset and a publicly accessible one, we demonstrate that the proposed approach consistently outperforms state-of-the-art methods based on generative adversarial networks and a CopyPaste strategy. Our proposed model demonstrates superior performance in standard distortion metrics and preserves critical ECG characteristics, particularly the P, Q, R, S, and T wave coordinates. Two emerging clinical applications emphasize the relevance of our work. The first is the increasing need to digitize paper-stored ECGs for utilization in AI-based applications (automatic annotation and risk-quantification), often limited to digital ECG complete 10s recordings. The second is the widespread use of wearable devices that record ECGs but typically capture only a small subset of the 12 standard leads. In both cases, a non-negligible amount of information is lost or not recorded, which our approach aims to recover to overcome these limitations.

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schr\"odinger type operators of the form $(-\Delta)^s +V$ in the unit ball $B$ in $\mathbb{R}^N$ with a nondecreasing radial potential $V$. Specifically, we show that the eigenspace corresponding to the second radial eigenvalue is simple and spanned by an eigenfunction $u$ which changes sign precisely once in the radial variable and does not have zeroes anywhere else in $B$. Moreover, by a new Hopf type lemma for supersolutions to a class of degenerate mixed boundary value problems, we show that $u$ has a nonvanishing fractional boundary derivative on $\partial B$. We apply this result to prove uniqueness and nondegeneracy of positive ground state solutions to the problem $(-\Delta)^s u+\lambda u=u^p$ on ${B}$, $\; u=0$ on $\mathbb{R}^N\setminus B$. Here $s\in (0,1)$, $\lambda\geq 0$ and $p>1$ is strictly smaller than the critical Sobolev exponent., Comment: 22 pages

Content and image generation consist in creating or generating data from noisy information by extracting specific features such as texture, edges, and other thin image structures. We are interested here in generative models, and two main problems are addressed. Firstly, the improvements of specific feature extraction while accounting at multiscale levels intrinsic geometric features; and secondly, the equivariance of the network to reduce its complexity and provide a geometric interpretability. To proceed, we propose a geometric generative model based on an equivariant partial differential equation (PDE) for group convolution neural networks (G-CNNs), so called PDE-G-CNNs, built on morphology operators and generative adversarial networks (GANs). Equivariant morphological PDE layers are composed of multiscale dilations and erosions formulated in Riemannian manifolds, while group symmetries are defined on a Lie group. We take advantage of the Lie group structure to properly integrate the equivariance in layers, and are able to use the Riemannian metric to solve the multiscale morphological operations. Each point of the Lie group is associated with a unique point in the manifold, which helps us derive a metric on the Riemannian manifold from a tensor field invariant under the Lie group so that the induced metric has the same symmetries. The proposed geometric morphological GAN (GM-GAN) is obtained by using the proposed morphological equivariant convolutions in PDE-G-CNNs to bring nonlinearity in classical CNNs. GM-GAN is evaluated on MNIST data and compared with GANs. Preliminary results show that GM-GAN model outperforms classical GAN.

This paper introduces a Bayesian framework for image inversion by deriving a probabilistic counterpart to the regularization-by-denoising (RED) paradigm. It additionally implements a Monte Carlo algorithm specifically tailored for sampling from the resulting posterior distribution, based on an asymptotically exact data augmentation (AXDA). The proposed algorithm is an approximate instance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo step. The proposed method is applied to common imaging tasks such as deblurring, inpainting and super-resolution, demonstrating its efficacy through extensive numerical experiments. These contributions advance Bayesian inference in imaging by leveraging data-driven regularization strategies within a probabilistic framework.

We prove that positive solutions $u\in H^s(\mathbb{R}^N)$ to the equation $(-\Delta )^s u+ u=u^p$ in $\mathbb{R}^N$ are nonradially nondegenerate, for all $s\in (0,1)$, $N\geq 1$ and $p>1$ strictly smaller than the critical Sobolev exponent. By this we mean that the linearized equation $(-\Delta )^s w+ w-pu^{p-1}w = 0$ does not admit nonradial solutions beside the directional derivatives of $u$. Letting $B$ be the unit centered ball and $\lambda_1(B)$ the first Dirichlet eigenvalue of the fractional Laplacian $(-\Delta )^s$, we also prove that positive solutions to $(-\Delta )^s u+\lambda u=u^p$ in ${B}$ with $u=0$ on $\mathbb{R}^N\setminus B$, are nonradially nondegenerate for any $\lambda> -\lambda_1(B)$ in the sense that the linearized equation does not admit nonradial solutions. From these results, we then deduce uniqueness and full nondegeneracy of positive solutions in some special cases. In particular, in the case $N=1$, we prove that the equation $(-\Delta )^s u+ u=u^2$ in $\mathbb{R}$ or in $B$, with zero exterior data, admits a unique even solution which is fully nondegenerate in the optimal range $s \in (\frac{1}{6},1)$, thus extending the classical uniqueness result of Amick and Toland on the Benjamin-Ono equation. Moreover, in the case $N=1$, $\lambda=0$, we also prove the uniqueness and full nondegeneracy of positive solutions for the Dirichlet problem in $B$ with arbitrary subcritical exponent $p$. Finally, we determine the unique positive ground state solution of $(-\Delta )^{\frac{1}{2}} u+ u=u^{p}$ in $\mathbb{R}^N$, $N \ge 1$ with $p=1+\frac{2}{N+1}$ and compute the sharp constant in the associated Gagliardo-Nirenberg inequality $$ \|u\|_{L^{p+1}(\mathbb{R}^N)} \le C \|(-\Delta )^{\frac{1}{4}} u\|_{L^2(\mathbb{R}^N)}^{\frac{N}{N+2}} \|u\|_{L^2(\mathbb{R}^N)}^{\frac{2}{N+2}}. $$, Comment: 32 pages. The first and second uploaded version contained a mistake, therefore the main results of the work were modified and the proofs were corrected in the third uploaded version on April 5, 2024. In this revised version we improve our previous results and prove uniqueness and nondegeneracy results in some special cases