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We consider the gravity-capillary water waves equations of a 2D fluid with constant vorticity. Using variational methods we prove the bifurcation of steady periodic traveling water waves for {\it all} the values of gravity, surface tension, constant vorticity, depth and wavelenght, extending previous results valid for restricted values of the parameters. We parametrize the bifurcating Stokes waves either with their speed or their momentum.

We overview the recent result [3, Theorem 1.1] about the high-frequency instability of Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of {\it isolas} parameterized by an integer $ \mathtt{p} \geq 2 $ for any value of the depth $ \mathtt{h} > 0 $ such that an explicit analytic function $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not zero. In [3] it is proved that the map $ \mathtt{h} \mapsto \beta_1^{(\mathtt{p})}(\mathtt{h}) $ is not identically zero for any $ \mathtt{p} \geq 2 $ by showing that $ \lim_{\mathtt{h} \to 0^+}\beta_1^{(\mathtt{p})}(\mathtt{h}) = - \infty $. In this manuscript we compute the asymptotic expansion of $\beta_1^{(\mathtt{p})}(\mathtt{h}) $ in the deep-water limit $ \mathtt{h} \to + \infty $ -- it vanishes exponentially fast to zero -- for $\mathtt{p}=2$, $3$, $4$., Comment: Supported by PRIN 2020 (2020XB3EFL001), PRIN 2022 (2022HSSYPN), ERC Stg 2021 HamDyWWa (101039762), ERC Cog 2023 GUnDHam (101124921). arXiv admin note: substantial text overlap with arXiv:2405.05854

Purposes: This study aimed to develop a computed tomography (CT)-based multi-organ segmentation model for delineating organs-at-risk (OARs) in pediatric upper abdominal tumors and evaluate its robustness across multiple datasets. Materials and methods: In-house postoperative CTs from pediatric patients with renal tumors and neuroblastoma (n=189) and a public dataset (n=189) with CTs covering thoracoabdominal regions were used. Seventeen OARs were delineated: nine by clinicians (Type 1) and eight using TotalSegmentator (Type 2). Auto-segmentation models were trained using in-house (ModelPMC-UMCU) and a combined dataset of public data (Model-Combined). Performance was assessed with Dice Similarity Coefficient (DSC), 95% Hausdorff Distance (HD95), and mean surface distance (MSD). Two clinicians rated clinical acceptability on a 5-point Likert scale across 15 patient contours. Model robustness was evaluated against sex, age, intravenous contrast, and tumor type. Results: Model-PMC-UMCU achieved mean DSC values above 0.95 for five of nine OARs, while spleen and heart ranged between 0.90 and 0.95. The stomach-bowel and pancreas exhibited DSC values below 0.90. Model-Combined demonstrated improved robustness across both datasets. Clinical evaluation revealed good usability, with both clinicians rating six of nine Type 1 OARs above four and six of eight Type 2 OARs above three. Significant performance 2 differences were only found across age groups in both datasets, specifically in the left lung and pancreas. The 0-2 age group showed the lowest performance. Conclusion: A multi-organ segmentation model was developed, showcasing enhanced robustness when trained on combined datasets. This model is suitable for various OARs and can be applied to multiple datasets in clinical settings., Comment: 23 pages, 5 figures, 1 table. Submitted to Radiotherapy and Oncology (2024-11-01)

We consider time dependently perturbed quantum harmonic oscillators in $\mathbb{R}^2$: $$ {\rm i} \partial_t u=\frac12(-\partial_{x_1}^2-\partial_{x_2}^2 + x_1^2+x_2^2)u +V(t, x, D)u, \qquad \ x \in \mathbb{R}^2, $$ where $V(t, x, D)$ is a selfadjoint pseudodifferential operator of degree zero, $2\pi$ periodic in time. We identify sufficient conditions on the principal symbol of the potential $V(t, x, D)$ that ensure existence of solutions exhibiting unbounded growth in time of their positive Sobolev norms and we show that the class of symbols satisfying such conditions is generic in the Fr\'echet space of classical $2\pi$- time periodic symbols of order zero. To prove our result we apply the abstract Theorem of arXiv:2101.09055v1 : the main difficulty is to find a conjugate operator $A$ for the resonant average of $V(t,x, D)$. We construct explicitly the symbol of the conjugate operator $A$, called escape function, combining techniques from microlocal analysis, dynamical systems and contact topology.

The exceptional magnetic, optical and phononic properties of Yttrium Iron Garnet (YIG) make it unique for spin-wave based and photonic applications. Yet, nanostructuring crystalline YIG and manipulating its magnetism in a non-destructive way is an outstanding challenge, and so far mostly limited to two-dimensional capabilities. Here, we show that irradiation of single-crystal YIG films with a focused UV laser drives a stable, giant enhancement of the perpendicular magnetic anisotropy, preserving the crystalline quality. This modulation is highly confined at the nanoscale in both the lateral and vertical directions, and its extension within the volume can be finely tuned with a continuous depth-control. By harnessing these three-dimensional anisotropy profiles, we demonstrate a large tuning of the spin-wave band structure, volume spatial localization, and non-reciprocity, realizing proof-of-principle 3D magnonic crystals. This straightforward, single-step, laser nanofabrication of three-dimensional magnetic systems based on crystalline YIG thin films opens the way to design novel functions in magnonic and magneto-optic devices., Comment: 17 pages, 5 figures

We consider the problem of transfer of energy to high frequencies in a quasilinear Schr\"odinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory.

Integration of Samarium Cobalt hard magnets on silicon requires buffer/protective layers that can enhance the magnetic properties of the magnet while preserving its structure and chemical composition after post-annealing treatments needed for the formation of the magnetically hard phase. In this work, a comparison of Samarium-Cobalt films for five different buffer/protective layers, namely Ti, W, TiW, Ta, Cr and two different annealing temperatures, 650{\deg}C and 750{\deg}C, is presented. Depending on materials and annealing temperatures, magnetic properties such as saturation and coercivity of the SmCo film can be finely tuned. We show that coercivity up to 3.65 T or saturation magnetization up to 0.95 T can be reached by proper choice of the relevant process parameters: deposition temperature, material for the buffer/protective layer and annealing temperature. Such value of coercivity is among the highest found in literature for thin films of SmCo.

Manual segmentation of medical images is labor intensive and especially challenging for images with poor contrast or resolution. The presence of disease exacerbates this further, increasing the need for an automated solution. To this extent, SynthSeg is a robust deep learning model designed for automatic brain segmentation across various contrasts and resolutions. This study validates the SynthSeg robust brain segmentation model on computed tomography (CT), using a multi-center dataset. An open access dataset of 260 paired CT and magnetic resonance imaging (MRI) from radiotherapy patients treated in 5 centers was collected. Brain segmentations from CT and MRI were obtained with SynthSeg model, a component of the Freesurfer imaging suite. These segmentations were compared and evaluated using Dice scores and Hausdorff 95 distance (HD95), treating MRI-based segmentations as the ground truth. Brain regions that failed to meet performance criteria were excluded based on automated quality control (QC) scores. Dice scores indicate a median overlap of 0.76 (IQR: 0.65-0.83). The median HD95 is 2.95 mm (IQR: 1.73-5.39). QC score based thresholding improves median dice by 0.1 and median HD95 by 0.05mm. Morphological differences related to sex and age, as detected by MRI, were also replicated with CT, with an approximate 17% difference between the CT and MRI results for sex and 10% difference between the results for age. SynthSeg can be utilized for CT-based automatic brain segmentation, but only in applications where precision is not essential. CT performance is lower than MRI based on the integrated QC scores, but low-quality segmentations can be excluded with QC-based thresholding. Additionally, performing CT-based neuroanatomical studies is encouraged, as the results show correlations in sex- and age-based analyses similar to those found with MRI., Comment: 15 pages, 9 figures, 3 supplementary data csv's, 1 supplementary file with 1 figure

One of the most appealing features of magnonics is the easy tunability of spin-waves propagation via external magnetic fields. Usually this requires bulky and power-hungry electromagnets which are not compatible with device miniaturization. Here we propose a different approach, exploiting the stray field from permanent micromagnets integrated on the same chip of a magnonic wave-guide. In our monolithic device, we employ two SmCo square micromagnets (10x10 $\mu$m$^2$) flanking a CoFeB conduit at different distances from its axis, to produce a tunable transverse bias field between 7.5 and 3.0 mT in the conduit region between the magnets. Spin waves excited by an antenna just outside the region between the magnets enter a region with a variable higher (lower) effective field when an external bias field is applied parallel (antiparallel) to that from the micromagnets. Consequently, the attenuation length and phase shift of Damon-Eshbach spin waves can be tuned in a wide range by playing with the parallel-antiparallel configuration of the external bias and the distance between SmCo micromagnets and the CoFeB conduit. This work demonstrates the potential of permanent micro-magnets for the realization of low-power, integrated magnonic devices with tunable functionalities.

We prove the long-standing conjecture regarding the existence of infinitely many high-frequency modulational instability ``isolas" for a Stokes wave in arbitrary depth $ \mathtt{h} > 0 $, subject to longitudinal perturbations. We completely describe the spectral bands with non-zero real part away from the origin of the $L^2(\mathbb{R})$-spectrum of the water waves system linearized at a Stokes waves of small amplitude $ \epsilon > 0 $. The unstable spectrum is the union of isolas of elliptical shape, parameterized by integers $ \mathtt{p}\geq 2 $, with semiaxis of size $ |\beta_1^{(\mathtt{p})} (\mathtt{h})| \epsilon^\mathtt{p}+ O(\epsilon^{\mathtt{p}+1} )$ where $\beta_1^{( \mathtt{p})} (\mathtt{h})$ is a nonzero analytic function of the depth $ \mathtt{h} $ that depends on the Taylor coefficients of the Stokes waves up to order $\mathtt{p}$., Comment: 58 pages, 10 figures, companion material at https://git-scm.sissa.it/amaspero/isolas/

In this paper we consider a family of generalized Korteweg-de Vries equations and study the linear modulational instability of small amplitude traveling waves solutions. Under explicit non-degeneracy conditions on the dispersion relation, we completely describe the spectrum near the origin of the linearized operator at such solutions and prove that the unstable spectrum (when present) is composed by branches depicting always a closed figure ''8''. We apply our abstract theorem to several equations such as the Whitham, the gravity-capillary Whitham and the Kawahara equations, confirming that the unstable spectrum of the corresponding linearized operators exhibits a figure ''8'' instability, as it was observed before only numerically. Our method of proof uses a symplectic version of Kato's theory of similarity transformation to reduce the problem to determine the eigenvalues of a $3 \times 3$ complex Hamiltonian and reversible matrix. Then, via a block-diagonalization procedure, we conjugate such matrix into a block-diagonal one composed by a $2\times 2$ Hamiltonian and reversible matrix, describing the unstable spectrum, and a single purely imaginary element describing the stable eigenvalue.

Radiation therapy plays a crucial role in cancer treatment, necessitating precise delivery of radiation to tumors while sparing healthy tissues over multiple days. Computed tomography (CT) is integral for treatment planning, offering electron density data crucial for accurate dose calculations. However, accurately representing patient anatomy is challenging, especially in adaptive radiotherapy, where CT is not acquired daily. Magnetic resonance imaging (MRI) provides superior soft-tissue contrast. Still, it lacks electron density information while cone beam CT (CBCT) lacks direct electron density calibration and is mainly used for patient positioning. Adopting MRI-only or CBCT-based adaptive radiotherapy eliminates the need for CT planning but presents challenges. Synthetic CT (sCT) generation techniques aim to address these challenges by using image synthesis to bridge the gap between MRI, CBCT, and CT. The SynthRAD2023 challenge was organized to compare synthetic CT generation methods using multi-center ground truth data from 1080 patients, divided into two tasks: 1) MRI-to-CT and 2) CBCT-to-CT. The evaluation included image similarity and dose-based metrics from proton and photon plans. The challenge attracted significant participation, with 617 registrations and 22/17 valid submissions for tasks 1/2. Top-performing teams achieved high structural similarity indices (>0.87/0.90) and gamma pass rates for photon (>98.1%/99.0%) and proton (>97.3%/97.0%) plans. However, no significant correlation was found between image similarity metrics and dose accuracy, emphasizing the need for dose evaluation when assessing the clinical applicability of sCT. SynthRAD2023 facilitated the investigation and benchmarking of sCT generation techniques, providing insights for developing MRI-only and CBCT-based adaptive radiotherapy., Comment: Preprint submitted to Medical Image Analysis

The fabrication and integration of high-quality structures of Yttrium Iron Garnet (YIG) is critical for magnonics.Films with excellent properties are obtained only on single crystal Gadolinium Gallium Garnet (GGG) substrates using high-temperature processes. The subsequent realization of magnonic structures via lithography and etching is not straightforward as it requires a tight control of the edge roughness, to avoid magnon scattering, and planarization in case of multilayer devices. In this work we describe a different approach based on local laser annealing of amorphous YIG films, avoiding the need for subjecting the entire sample to high thermal budgets and for physical etching. Starting from amorphous and paramagnetic YIG films grown by pulsed laser deposition at room temperature on GGG, a 405 nm laser is used for patterning arbitrary shaped ferrimagnetic structures by local crystallization. In thick films (160 nm) the laser induced surface corrugation prevents the propagation of spin-wave modes in patterned conduits. For thinner films (80 nm) coherent propagation is observed in 1.2 micron wide conduits displaying an attenuation length of 5 micron which is compatible with a damping coefficient of about 5e-3. Possible routes to achieve damping coefficients compatible with state-of-the art epitaxial YIG films are discussed., Comment: 29 pages, 6 figures, This is the pre-peer reviewed version of the following article: "Patterning magnonic structures via laser induced crystallization of Yittrium Iron Garnet", by A. Del Giacco et al. https://doi.org/10.1002/adfm.202401129, which has been accepted for publication in Advanced Functional Materials

We consider a modulated magnetic field, $B(t) = B_0 +\varepsilon f(\omega t)$, perpendicular to a fixed plane, where $B_0$ is constant, $\varepsilon>0$ and $f$ a periodic function on the torus ${\mathbb T}^n$. Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for "almost all" non resonant frequencies $\omega$. On the contrary, for the symmetric gauge we obtain that, for "almost all" non resonant frequencies $\omega$, the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.

We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from $\mathtt{i}\frac34$. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a $2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one., Comment: 35 pages, 4 figures, companion code https://git-scm.sissa.it/amaspero/first-isola-of-modulational-instability-of-stokes-waves-in-deep-water

This paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves -- called Stokes waves -- at the critical Whitham-Benjamin depth $ \mathtt{h}_{\scriptscriptstyle WB} = 1.363... $ and nearby values. We prove that Stokes waves of small amplitude $ \mathcal{O}( \epsilon ) $ are, at the critical depth $ \mathtt{h}_{\scriptscriptstyle WB} $, linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^2 $, $ c > 0 $, depending on the amplitude of the wave. This problem was not rigorously solved in previous literature because the expansions degenerate at the critical depth. In order to resolve this degenerate case, and describe in a mathematically exhaustive way how the eigenvalues change their stable-to-unstable nature along this shallow-to-deep water transient, we Taylor expand the computations of arXiv:2204.00809v2 at a higher degree of accuracy, derived by the fourth order expansion of the Stokes waves. We prove that also in this transient regime a pair of unstable eigenvalues depict a closed figure "8", of smaller size than for $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} $, as the Floquet exponent varies., Comment: 52 pages, 6 figures, companion Mathematica code available at https://git-scm.sissa.it/amaspero/benjamin-feir-instability. arXiv admin note: text overlap with arXiv:2204.00809

Inspired by recent graded metamaterials designs, we create phononic arrays of micro-resonators for frequency signal amplification and wave filtering. Leveraging suspended waveguides on a thick silicon substrate, we hybridize surface Rayleigh and Lamb flexural waves to effectively achieve phononic signal control along predefined channels. The guided waves are then spatially controlled using a suitable grading of the micro-resonators, which provide high signal-to-noise ratio and simultaneously create phononic delay-lines. The proposed device can be used for sensing, wave filtering or energy harvesting.

In this study, we develop a physics-informed deep learning-based method to synthesize multiple brain magnetic resonance imaging (MRI) contrasts from a single five-minute acquisition and investigate its ability to generalize to arbitrary contrasts to accelerate neuroimaging protocols. A dataset of fifty-five subjects acquired with a standard MRI protocol and a five-minute transient-state sequence was used to develop a physics-informed deep learning-based method. The model, based on a generative adversarial network, maps data acquired from the five-minute scan to "effective" quantitative parameter maps, here named q*-maps, by using its generated PD, T1, and T2 values in a signal model to synthesize four standard contrasts (proton density-weighted, T1-weighted, T2-weighted, and T2-weighted fluid-attenuated inversion recovery), from which losses are computed. The q*-maps are compared to literature values and the synthetic contrasts are compared to an end-to-end deep learning-based method proposed by literature. The generalizability of the proposed method is investigated for five volunteers by synthesizing three non-standard contrasts unseen during training and comparing these to respective ground truth acquisitions via contrast-to-noise ratio and quantitative assessment. The physics-informed method was able to match the high-quality synthMRI of the end-to-end method for the four standard contrasts, with mean \pm standard deviation structural similarity metrics above 0.75 \pm 0.08 and peak signal-to-noise ratios above 22.4 \pm 1.9 and 22.6 \pm 2.1. Additionally, the physics-informed method provided retrospective contrast adjustment, with visually similar signal contrast and comparable contrast-to-noise ratios to the ground truth acquisitions for three sequences unused for model training, demonstrating its generalizability and potential application to accelerate neuroimaging protocols., Comment: 23 pages, 7 figures, 1 table. Presented at ISMRM 2022. Will be submitted to NMR in biomedicine

Purpose: Medical imaging has become increasingly important in diagnosing and treating oncological patients, particularly in radiotherapy. Recent advances in synthetic computed tomography (sCT) generation have increased interest in public challenges to provide data and evaluation metrics for comparing different approaches openly. This paper describes a dataset of brain and pelvis computed tomography (CT) images with rigidly registered CBCT and MRI images to facilitate the development and evaluation of sCT generation for radiotherapy planning. Acquisition and validation methods: The dataset consists of CT, CBCT, and MRI of 540 brains and 540 pelvic radiotherapy patients from three Dutch university medical centers. Subjects' ages ranged from 3 to 93 years, with a mean age of 60. Various scanner models and acquisition settings were used across patients from the three data-providing centers. Details are available in CSV files provided with the datasets. Data format and usage notes: The data is available on Zenodo (https://doi.org/10.5281/zenodo.7260705) under the SynthRAD2023 collection. The images for each subject are available in nifti format. Potential applications: This dataset will enable the evaluation and development of image synthesis algorithms for radiotherapy purposes on a realistic multi-center dataset with varying acquisition protocols. Synthetic CT generation has numerous applications in radiation therapy, including diagnosis, treatment planning, treatment monitoring, and surgical planning., Comment: 15 pages, 4 figures, 9 tables, pre-print submitted to Medical Physics - dataset. The training dataset is available on Zenodo at https://doi.org/10.5281/zenodo.7260705 from April, 1st 2023

Background: Synthetic computed tomography (sCT) has been proposed and increasingly clinically adopted to enable magnetic resonance imaging (MRI)-based radiotherapy. Deep learning (DL) has recently demonstrated the ability to generate accurate sCT from fixed MRI acquisitions. However, MRI protocols may change over time or differ between centres resulting in low-quality sCT due to poor model generalisation. Purpose: investigating domain randomisation (DR) to increase the generalisation of a DL model for brain sCT generation. Methods: CT and corresponding T1-weighted MRI with/without contrast, T2-weighted, and FLAIR MRI from 95 patients undergoing RT were collected, considering FLAIR the unseen sequence where to investigate generalisation. A ``Baseline'' generative adversarial network was trained with/without the FLAIR sequence to test how a model performs without DR. Image similarity and accuracy of sCT-based dose plans were assessed against CT to select the best-performing DR approach against the Baseline. Results: The Baseline model had the poorest performance on FLAIR, with mean absolute error (MAE)=106$\pm$20.7 HU (mean$\pm\sigma$). Performance on FLAIR significantly improved for the DR model with MAE=99.0$\pm$14.9 HU, but still inferior to the performance of the Baseline+FLAIR model (MAE=72.6$\pm$10.1 HU). Similarly, an improvement in $\gamma$-pass rate was obtained for DR vs Baseline. Conclusions: DR improved image similarity and dose accuracy on the unseen sequence compared to training only on acquired MRI. DR makes the model more robust, reducing the need for re-training when applying a model on sequences unseen and unavailable for retraining., Comment: Preprint submitted to Physica Medica on 2023-02-16 for review. Also published in Zenodo at https://doi.org/10.5281/zenodo.7742642

We prove an almost global in time existence result of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that usual paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) does not preserve the Hamiltonian structure. A major novelty of this paper is to develop an algorithmic perturbative procedure \`a la Darboux to correct usual paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. The symplectic correctors turn out to be smoothing perturbations of the identity, and therefore only slightly modify the paradifferential structure of the equations. The Darboux procedure which recovers the nonlinear Hamiltonian structure is written in an abstract functional setting, in order to be applicable also in other contexts., Comment: 132 pages

While roton dispersion relations had been restricted to correlated quantum systems at low temperature, recent works show the possibility of obtaining this unusual dispersion in acoustic and elastic metamaterials. Such phenomenon has been demonstrated in periodic structures by means of beyond-nearest-neighbor interactions, following the formulation firstly developed by Brillouin in the $'50s$. In this paper, we demonstrate both numerically and experimentally that beyond-nearest-neighbor connections are not a necessary condition to obtain this unusual dispersion relation in elasticity. Leveraging the intrinsic complexity of elastic systems supporting different types of waves, we demonstrate that mode locking can be applied to obtain roton dispersion, without the need of elastic or magnetic interactions between non nearest neighbors. Moreover, the combination of roton dispersion and rainbow physics enables spatial separation of the energy fluxes with positive and negative group velocity.

Background: Respiratory-resolved four-dimensional magnetic resonance imaging (4D-MRI) provides essential motion information for accurate radiation treatments of mobile tumors. However, obtaining high-quality 4D-MRI suffers from long acquisition and reconstruction times. Purpose: To develop a deep learning architecture to quickly acquire and reconstruct high-quality 4D-MRI, enabling accurate motion quantification for MRI-guided radiotherapy. Methods: A small convolutional neural network called MODEST is proposed to reconstruct 4D-MRI by performing a spatial and temporal decomposition, omitting the need for 4D convolutions to use all the spatio-temporal information present in 4D-MRI. This network is trained on undersampled 4D-MRI after respiratory binning to reconstruct high-quality 4D-MRI obtained by compressed sensing reconstruction. The network is trained, validated, and tested on 4D-MRI of 28 lung cancer patients acquired with a T1-weighted golden-angle radial stack-of-stars sequence. The 4D-MRI of 18, 5, and 5 patients were used for training, validation, and testing. Network performances are evaluated on image quality measured by the structural similarity index (SSIM) and motion consistency by comparing the position of the lung-liver interface on undersampled 4D-MRI before and after respiratory binning. The network is compared to conventional architectures such as a U-Net, which has 30 times more trainable parameters. Results: MODEST can reconstruct high-quality 4D-MRI with higher image quality than a U-Net, despite a thirty-fold reduction in trainable parameters. High-quality 4D-MRI can be obtained using MODEST in approximately 2.5 minutes, including acquisition, processing, and reconstruction. Conclusion: High-quality accelerated 4D-MRI can be obtained using MODEST, which is particularly interesting for MRI-guided radiotherapy., Comment: Code available at https://gitlab.com/computational-imaging-lab/modest

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth $\mathtt h$ is larger than a critical threshold $\mathtt h_{WB} \approx 1.363$. In this paper we completely describe, for any value of $\mathtt h > 0$, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent $\mu$ is turned on. We prove in particular the existence of a unique depth $\mathtt h_{WB}$, which coincides with the one predicted by Whitham and Benjamin, such that, for any $0 < \mathtt h < \mathtt h_{WB}$, the eigenvalues close to zero remain purely imaginary and, for any $\mathtt h > \mathtt h_{WB}$, a pair of non-purely imaginary eigenvalues depicts a closed figure "8", parameterized by the Floquet exponent. As $\mathtt h \to \mathtt h_{WB}^+$ this figure "8" collapses to the origin of the complex plane. The proof combines a symplectic version of Kato's perturbative theory to compute the eigenvalues of a $4 \times 4$ Hamiltonian and reversible matrix, and KAM inspired transformations to block-diagonalize it. The four eigenvalues have all the same size $O(\mu)$ - unlike the infinitely deep water case in [6]- and the correct Benjamin-Feir phenomenon appears only after one non-perturbative block-diagonalization step. In addition one has to track, along the whole proof, the explicit dependence of the entries of the $4 \times 4$ reduced matrix with respect to the depth $\mathtt h$., Comment: 47 pages, 5 figures, revised introduction. arXiv admin note: substantial text overlap with arXiv:2109.11852