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We consider the relativistic Vlasov-Maxwell system in three dimensions and study the limiting asymptotic behavior as $t \to \infty$ of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly-supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the limiting charge density does not vanish, our estimates are sharp up to a logarithmic correction. However, when this quantity is identically zero in the limit, the limiting current density and fields may also vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system., Comment: 49 pages, 2 figures

This work is concerned with operators of the type A = --c$\Delta$ acting in domains $\Omega$ := $\Omega$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $\epsilon$ > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $\epsilon$ $\rightarrow$ 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of ''layered media'' enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y) $\in$ C 2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of ''flattening out'' of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a ''minimal amplitude hypothesis''. However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.

We introduce a novel approach for gait transfer from unconstrained videos in-the-wild. In contrast to motion transfer, the objective here is not to imitate the source's motions by the target, but rather to replace the walking source with the target, while transferring the target's typical gait. Our approach can be trained only once with multiple sources and is able to transfer the gait of the target from unseen sources, eliminating the need for retraining for each new source independently. Furthermore, we propose a novel metrics for gait transfer based on gait recognition models that enable to quantify the quality of the transferred gait, and show that existing techniques yield a discrepancy that can be easily detected. We introduce Cycle Transformers GAN (CTrGAN), that consist of a decoder and encoder, both Transformers, where the attention is on the temporal domain between complete images rather than the spatial domain between patches. Using a widely-used gait recognition dataset, we demonstrate that our approach is capable of producing over an order of magnitude more realistic personalized gaits than existing methods, even when used with sources that were not available during training. As part of our solution, we present a detector that determines whether a video is real or generated by our model.

Adaptive image restoration models can restore images with different degradation levels at inference time without the need to retrain the model. We present an approach that is highly accurate and allows a significant reduction in the number of parameters. In contrast to existing methods, our approach can restore images using a single fixed-size model, regardless of the number of degradation levels. On popular datasets, our approach yields state-of-the-art results in terms of size and accuracy for a variety of image restoration tasks, including denoising, deJPEG, and super-resolution., Comment: 5 pages, 5 Figures, ICASSP 2023

This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm require to return the potential and boundary conditions? It is shown that if all but finitely many of the eigenvalues and norming constants coincide with those for the zero potential then the number of limits is zero, i.e. it is possible to retrieve the potential and boundary conditions precisely in finitely many steps. Otherwise, it is shown that this problem requires a single limit; moreover, if one has a priori control over how much the eigenvalues and norming constants differ from those of the zero-potential problem, and one knows that the average of the potential is zero, then the computation can be performed with complete error control. This is done in the spirit of the Solvability Complexity Index. All algorithms are provided explicitly along with numerical examples., Comment: 27 pages

We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $t \to \infty$ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $t \to \infty$ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may "forget" any previous configuration of particles., Comment: 27 pages

The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified ``toy model'' is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation., Comment: 13 pages, final version for publication in KRM

Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a "one-size-fits-all" algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schr\"odinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schr\"odinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits., Comment: 32 pages, 6 figures

In this paper we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq3$. In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric, and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds., Comment: 38 pages

Existing methods for enhancing dark images captured in a very low-light environment assume that the intensity level of the optimal output image is known and already included in the training set. However, this assumption often does not hold, leading to output images that contain visual imperfections such as dark regions or low contrast. To facilitate the training and evaluation of adaptive models that can overcome this limitation, we have created a dataset of 1500 raw images taken in both indoor and outdoor low-light conditions. Based on our dataset, we introduce a deep learning model capable of enhancing input images with a wide range of intensity levels at runtime, including ones that are not seen during training. Our experimental results demonstrate that our proposed dataset combined with our model can consistently and effectively enhance images across a wide range of diverse and challenging scenarios.

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $\mathcal C^2$. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point., Comment: 25 pages, 12 figures

Consider a conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the Klein-Gordon equations with inverse-square potentials in the space $X$, proving in particular global-in-time Strichartz estimates in this setting., Comment: 44 pages. Comments are welcome

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the "meaningful objects" are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary., Comment: 8 pages. arXiv admin note: text overlap with arXiv:1902.09047

We present a novel approach for RANSAC-based computation of the fundamental matrix based on epipolar homography decomposition. We analyze the geometrical meaning of the decomposition-based representation and show that it directly induces a consecutive sampling strategy of two independent sets of correspondences. We show that our method guarantees a minimal number of evaluated hypotheses with respect to current minimal approaches, on the condition that there are four correspondences on an image line. We validate our approach on real-world image pairs, providing fast and accurate results.

The question of whether it is possible to compute scattering resonances of Schr\"odinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be $\mathcal{C}^1$. The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested point., Comment: 19 pages, 6 figures

Wavelet denoising is a classical and effective approach for reducing noise in images and signals. Suggested in 1994, this approach is carried out by rectifying the coefficients of a noisy image in the transform domain, using a set of scalar shrinkage function (SFs). A plethora of papers deals with the optimal shape of the SFs and the transform used, where it is known that applying the SFs in redundant bases provides improved results. This paper provides a complete picture of the interrelations between the transform used, the optimal shrinkage functions, and the domains in which they are optimized. In particular, we show that for subband optimization, where each SF is optimized independently for a particular band, optimizing the SFs in the spatial domain is always better than or equal to optimizing the SFs in the transform domain. For redundant bases, we provide the expected denoising gain we may achieve, relative to the unitary basis, as a function of the redundancy rate.

Discrete approximations to the equation \begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A'(x)+H(x)) u^{(1)} + B(x) u = f, \; x\in[0,1] \end{equation*} are considered. This is an extension of the Sturm-Liouville case $D(x)\equiv H(x)\equiv 0$ [ M. Ben-Artzi, J.-P. Croisille, D. Fishelov and R. Katzir, Discrete fourth-order Sturm-Liouville problems, IMA J. Numer. Anal. {\bf 38} (2018), 1485-1522. doi: 10.1093/imanum/drx038] to the non-self-adjoint setting. The "natural" boundary conditions in the Sturm-Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second, third and fourth order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties--compactness and coercivity. It allows to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented., Comment: 4 figures

Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose another open set $\omega\subset \Omega$. The estimate $0C_{\omega}>0,\exists \omega, \omega\not=\emptyset$, such that $\inf R_{\omega}(u)=0$,and we wish to characterize these two sets. For two patterns we give a sufficient condition, sometimes necessary. As our operator corresponds to a layered media we can give another representation of its spectrum: i.e. a subset of points of $R\times R$ that leads to the suggested partition and others connected results: micro local interpretation, default measures,... Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added., Comment: in French, 34 pages. Rehandled version of arXiv:1911.09947 since the results of section 4.1 was false. This section becomes section 3.1 with new statement and proof. Others modifications: new Theorems, corollaries and lemmas and new numerotation. As a result, we have profoundly changed the Introduction

The one-dimensional viscous conservation law is considered on the whole line $$ u_t + f(u)_x=\eps u_{xx},\quad (x,t)\in\RR\times\overline{\RP},\quad \eps>0, $$ subject to positive measure initial data. The flux $f\in C^1(\RR)$ is assumed to satisfy a $p-$condition, a weak form of convexity. Existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for viscous Hamilton-Jacobi equations.

This paper addresses the three concepts of \textit{ consistency, stability and convergence } in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of "balance laws". Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of "Godunov compatibility", which serves as a substitute to the entropy condition., Comment: 27 pages

Motivated by the well-known phase-space portrait of the nonlinear pendulum, the purpose of this paper is to obtain convergence rates in the ergodic theorem for flows in the plane that have arbitrarily slow trajectories. Considering bounded periodic trajectories near the heteroclinic orbits, it is shown that despite lacking a spectral gap, there exists a functional space (which is a strict subset of $L^2$) on which time averages converge uniformly to spatial averages (with an explicit rate). The main ingredient of the proof is an estimate of the density of the spectrum of the generator of the flow near zero., Comment: 13 pages, completely revised

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schr\"odinger and wave equations. In particular, decay estimates for time-averages of solutions are shown., Comment: 13 pages

Let $$L_0=\suml_{j=1}^nM_j^0D_j+M_0^0,\,\,\,\,D_j=\frac{1}{i}\frac{\pa}{\paxj}, \quad x\in\Rn,$$ be a constant coefficient first-order partial differential system, where the matrices $M_j^0$ are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the nonhomegeneous case it is assumed that the operator is isotropic . The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: \begin{itemize} \item Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular the Dirac and Maxwell systems are explicitly treated. \item The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation. \end{itemize}