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We demonstrate that a ReLU deep neural network with a width of $2$ and a depth of $2N+4M-1$ layers can achieve finite sample memorization for any dataset comprising $N$ elements in $\mathbb{R}^d$, where $d\ge1,$ and $M$ classes, thereby ensuring accurate classification. By modeling the neural network as a time-discrete nonlinear dynamical system, we interpret the memorization property as a problem of simultaneous or ensemble controllability. This problem is addressed by constructing the network parameters inductively and explicitly, bypassing the need for training or solving any optimization problem. Additionally, we establish that such a network can achieve universal approximation in $L^p(\Omega;\mathbb{R}_+)$, where $\Omega$ is a bounded subset of $\mathbb{R}^d$ and $p\in[1,\infty)$, using a ReLU deep neural network with a width of $d+1$. We also provide depth estimates for approximating $W^{1,p}$ functions and width estimates for approximating $L^p(\Omega;\mathbb{R}^m)$ for $m\geq1$. Our proofs are constructive, offering explicit values for the biases and weights involved.

This paper investigates the application of mini-batch gradient descent to semiflows. Given a loss function, we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss functions over time, defining a piecewise flow. We prove that, under suitable assumptions on the gradient flow, the mini-batch descent flow trajectory closely approximates the original gradient flow trajectory on average. Additionally, we propose a randomized minimizing movement scheme that also approximates the gradient flow of the loss function. We illustrate the versatility of this approach across various problems, including constrained optimization, sparse inversion, and domain decomposition. Finally, we validate our results with several numerical examples., Comment: 25 pages, 14 figures

Our main contribution in this article is the achievement of the turnpike property in its integral and exponential forms for parameter-dependent systems with averaged observations in the cost functional. Namely, under suitable assumptions with respect to the matrices that defined the dynamics and the cost functional, we prove that the optimal control and state for the evolutionary problem converge in average to the optimal pair of an associated stationary problem. Moreover, we characterize the closeness between these two optimal solutions, proving that over a large time interval, they are exponentially close.

Ion acceleration from gaseous targets driven by relativistic-intensity lasers was demonstrated as early as the late 90s, yet most of the experiments conducted to date have involved picosecond-duration, Nd:glass lasers operating at low repetition rate. Here, we present measurements on the interaction of ultraintense ($\sim 10^{20}\,\rm W\,cm^{-2}$, 1 PW), ultrashort ($\sim 70\,\rm fs$) Ti:Sa laser pulses with near-critical ($\sim 10^{20}\,\rm cm^{-3}$) helium gas jets, a debris-free targetry compatible with high ($\sim 1\,\rm Hz$) repetition rate operation. We provide evidence of $\alpha$ particles being forward accelerated up to $\sim 2.7\,\rm MeV$ energy with a total flux of $\sim 10^{11}\,\rm sr^{-1}$ as integrated over $>0.1 \,\rm MeV$ energies and detected within a $0.5\,\rm mrad$ solid angle. We also report on on-axis emission of relativistic electrons with an exponentially decaying spectrum characterized by a $\sim 10\,\rm MeV$ slope, i.e., five times larger than the standard ponderomotive scaling. The total charge of these electrons with energy above 2 MeV is estimated to be of $\sim 1 \,\rm nC$, corresponding to $\sim 0.1\,\%$ of the laser drive energy. In addition, we observe the formation of a plasma channel, extending longitudinally across the gas density maximum and expanding radially with time. These results are well captured by large-scale particle-in-cell simulations, which reveal that the detected fast ions most likely originate from reflection off the rapidly expanding channel walls. The latter process is predicted to yield ion energies in the MeV range, which compare well with the measurements. Finally, direct laser acceleration is shown to be the dominant mechanism behind the observed electron energization.

Motivated by singular limits for long-time optimal control problems, we investigate a class of parameter-dependent parabolic equations. First, we prove a turnpike result, uniform with respect to the parameters within a suitable regularity class and under appropriate bounds. The main ingredient of our proof is the justification of the uniform exponential stabilization of the corresponding Riccati equations, which is derived from the uniform null control properties of the model. Then, we focus on a heat equation with rapidly oscillating coefficients. In the one-dimensional setting, we obtain a uniform turnpike property with respect to the highly oscillatory heterogeneous medium. Afterward, we establish the homogenization of the turnpike property. Finally, our results are validated by numerical experiments.

We present a Hubble Space Telescope (HST) weak gravitational lensing study of nine distant and massive galaxy clusters with redshifts $1.0 \lesssim z \lesssim 1.7$ ($z_\mathrm{median} = 1.4$) and Sunyaev Zel'dovich (SZ) detection significance $\xi > 6.0$ from the South Pole Telescope Sunyaev Zel'dovich (SPT-SZ) survey. We measured weak lensing galaxy shapes in HST/ACS F606W and F814W images and used additional observations from HST/WFC3 in F110W and VLT/FORS2 in $U_\mathrm{HIGH}$ to preferentially select background galaxies at $z\gtrsim 1.8$, achieving a high purity. We combined recent redshift estimates from the CANDELS/3D-HST and HUDF fields to infer an improved estimate of the source redshift distribution. We measured weak lensing masses by fitting the tangential reduced shear profiles with spherical Navarro-Frenk-White (NFW) models. We obtained the largest lensing mass in our sample for the cluster SPT-CLJ2040$-$4451, thereby confirming earlier results that suggest a high lensing mass of this cluster compared to X-ray and SZ mass measurements. Combining our weak lensing mass constraints with results obtained by previous studies for lower redshift clusters, we extended the calibration of the scaling relation between the unbiased SZ detection significance $\zeta$ and the cluster mass for the SPT-SZ survey out to higher redshifts. We found that the mass scale inferred from our highest redshift bin ($1.2 < z < 1.7$) is consistent with an extrapolation of constraints derived from lower redshifts, albeit with large statistical uncertainties. Thus, our results show a similar tendency as found in previous studies, where the cluster mass scale derived from the weak lensing data is lower than the mass scale expected in a Planck $\nu\Lambda$CDM (i.e. $\nu$ $\Lambda$ Cold Dark Matter) cosmology given the SPT-SZ cluster number counts., Comment: 23 pages, 10 figures + appendix (9 pages, 9 figures); accepted for publication in A&A

Expanding from previous work we present weak lensing measurements for a total sample of 30 distant ($z_\mathrm{median}=0.93$) massive galaxy clusters from the South Pole Telescope Sunyaev-Zel'dovich (SPT-SZ) Survey, measuring galaxy shapes in Hubble Space Telescope (HST) Advanced Camera for Surveys images. We remove cluster members and preferentially select $z\gtrsim 1.4$ background galaxies via $V-I$ colour, employing deep photometry from VLT/FORS2 and Gemini-South/GMOS. We apply revised calibrations for the weak lensing shape measurements and the source redshift distribution to estimate the cluster masses. In combination with earlier Magellan/Megacam results for lower-redshifts clusters we infer refined constraints on the scaling relation between the SZ detection significance and the cluster mass, in particular regarding its redshift evolution. The mass scale inferred from the weak lensing data is lower by a factor $0.76^{+0.10}_{-0.14}$ (at our pivot redshift $z=0.6$) compared to what would be needed to reconcile a flat Planck $\nu\Lambda$CDM cosmology (in which the sum of the neutrino masses is a free parameter) with the observed SPT-SZ cluster counts. In order to sensitively test the level of (dis-)agreement between SPT clusters and Planck, further expanded weak lensing follow-up samples are needed., Comment: 32 pages, 20 figures, 12 tables, matches the version accepted for publication in MNRAS

WL measurements have well-known shear estimation biases, which can be partially corrected for with the use of image simulations. We present an analysis of simulated images that mimic HST/ACS observations of high-redshift galaxy clusters, including cluster specific issues such as non-weak shear and increased blending. Our synthetic galaxies have been generated to match the observed properties of the background-selected samples in the real images. First, we used simulations with galaxies on a grid to determine a revised signal-to-noise-dependent correction for multiplicative shear measurement bias, and to quantify the sensitivity of our bias calibration to mismatches of galaxy or PSF properties between the real data and the simulations. We studied the impact of increased blending and light contamination from cluster and foreground galaxies, finding it negligible for $z>0.7$ clusters, whereas there is an effect at the $\sim 1\%$ level for lower redshift clusters. Finally, we studied the impact of fainter neighbours and selection bias mimicking the positions and magnitudes of galaxies in CANDELS data. The initial SExtractor object detection causes a selection bias of $-0.028 \pm 0.002$, reduced to $-0.016 \pm 0.002$ by further cuts. We compared our CANDELS-based estimate to a grid-based analysis, with added clustered galaxies reaching even fainter magnitudes, yielding a refined estimate of $\sim -0.013$. Our pipeline is calibrated to an accuracy of $\sim 0.015$, which is fully sufficient for current and near-future weak lensing studies of high-redshift clusters. As an application, we used it for a refined analysis of three highly relaxed clusters from the SPT-SZ survey, including measurements down to $r>200$ kpc. Compared to previously employed scales, this tightens the cluster mass constraints by a factor 1.38 on average., Comment: 21 pages, 19 figures. Matches the version accepted for publication in A&A

We introduce a fast spatial point pattern analysis technique which is suitable for systems of many identical particles giving rise to multi-particle correlations up to arbitrary order. The obtained correlation parameters allow to quantify the quality of mean field assumptions or theories that incorporate correlations of limited order. We study the Vicsek model of self-propelled particles and create a correlation map marking the required correlation order for each point in phase space incorporating up to ten-particle correlations. We find that multi-particle correlations are important even in a large part of the disordered phase. Furthermore, the two-particle correlation parameter serves as an excellent order parameter to locate both phase transitions of the system, whereas two different order parameters were required before., Comment: 19 pages, 18 figures

In modern weak-lensing surveys, the common approach to correct for residual systematic biases in the shear is to calibrate shape measurement algorithms using simulations. These simulations must fully capture the complexity of the observations to avoid introducing any additional bias. In this paper we study the importance of faint galaxies below the observational detection limit of a survey. We simulate simplified Euclid VIS images including and excluding this faint population, and measure the shift in the multiplicative shear bias between the two sets of simulations. We measure the shear with three different algorithms: a moment-based approach, model fitting, and machine learning. We find that for all methods, a spatially uniform random distribution of faint galaxies introduces a shear multiplicative bias of the order of a few times $10^{-3}$. This value increases to the order of $10^{-2}$ when including the clustering of the faint galaxies, as measured in the Hubble Space Telescope Ultra-Deep Field. The magnification of the faint background galaxies due to the brighter galaxies along the line of sight is found to have a negligible impact on the multiplicative bias. We conclude that the undetected galaxies must be included in the calibration simulations with proper clustering properties down to magnitude 28 in order to reach a residual uncertainty on the multiplicative shear bias calibration of a few times $10^{-4}$, in line with the $2\times10^{-3}$ total accuracy budget required by the scientific objectives of the Euclid survey. We propose two complementary methods for including faint galaxy clustering in the calibration simulations., Comment: Version published in A&A