Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials

We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials V,q\in H^r_{\per}(\R) for $r\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\ast$ such that for $p\in (1,p^\ast)$ both in the "+" and the "-" case we can use variational methods to prove existence of time-periodic real-valued solutions that are localized in the space direction. The potentials are constructed explicitely in class (P1) and (P2) and are found by a recent result from inverse spectral theory in class (P3). The critical exponent $p^\ast$ depends on the regularity of $V, q$. Our result builds upon a Fourier expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the potentials and the spatial and temporal periods, the spectrum of the wave operator $V(x)\partial_t^2-\partial_x^2+q(x)$ (considered on suitable space of time-periodic functions) is bounded away from $0$. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for strongly indefinite variational problems

Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient $V$ and subcritical cylindrically symmetric nonlinearity $f$. The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions, new rearrangement type inequalities from Brock and the recent extension of the Nehari-manifold technique by Szulkin and Weth.Comment: 13 page

Discriminative Transfer Learning for General Image Restoration

Recently, several discriminative learning approaches have been proposed for effective image restoration, achieving convincing trade-off between image quality and computational efficiency. However, these methods require separate training for each restoration task (e.g., denoising, deblurring, demosaicing) and problem condition (e.g., noise level of input images). This makes it time-consuming and difficult to encompass all tasks and conditions during training. In this paper, we propose a discriminative transfer learning method that incorporates formal proximal optimization and discriminative learning for general image restoration. The method requires a single-pass training and allows for reuse across various problems and conditions while achieving an efficiency comparable to previous discriminative approaches. Furthermore, after being trained, our model can be easily transferred to new likelihood terms to solve untrained tasks, or be combined with existing priors to further improve image restoration quality

Real-valued, time-periodicweak solutions for a semilinear wave equation with periodic δ-potential

We consider the semilinear wave equation $V(x)$$u$$_{u}$ − $u$$_{xx}$ = $±|u|$$^{p-1}$$u$ with p ∈ (1, $\frac{5}{3}$) and a periodically extended delta potential $V(x)$ = $α + βδper(x)$. Both the “+” and the “-” case can be treated. We prove the existence of time-periodic real-valued solutions that are localized in the space direction. Our result builds upon a Fourier-Floquet-Bloch expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the parameters α, β and the spatial and temporal periods, the spectrum of the wave operator $V(x)$∂$\frac{2}{t}$ - ∂$\frac{2}{x}$ (considered on suitable space of time-periodic functions) is bounded away from 0. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for indefinite variational problems

Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients

We consider the nonlinear curl-curl problem ∇ × ∇ × U + V(x)U = f (x, |U|2)U in R3 related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient V and subcritical cylindrically symmetric nonlinearity f . The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions [11, 12], new rearrangement type inequalities from [6] and the recent extension of the Nehari-manifold technique from [18]

Polarization fields: dynamic light field display using multi-layer LCDs

We introduce polarization field displays as an optically-efficient design for dynamic light field display using multi-layered LCDs. Such displays consist of a stacked set of liquid crystal panels with a single pair of crossed linear polarizers. Each layer is modeled as a spatially-controllable polarization rotator, as opposed to a conventional spatial light modulator that directly attenuates light. Color display is achieved using field sequential color illumination with monochromatic LCDs, mitigating severe attenuation and moiré occurring with layered color filter arrays. We demonstrate such displays can be controlled, at interactive refresh rates, by adopting the SART algorithm to tomographically solve for the optimal spatially-varying polarization state rotations applied by each layer. We validate our design by constructing a prototype using modified off-the-shelf panels. We demonstrate interactive display using a GPU-based SART implementation supporting both polarization-based and attenuation-based architectures. Experiments characterize the accuracy of our image formation model, verifying polarization field displays achieve increased brightness, higher resolution, and extended depth of field, as compared to existing automultiscopic display methods for dual-layer and multi-layer LCDs.National Science Foundation (U.S.) (Grant IIS-1116452)United States. Defense Advanced Research Projects Agency (Grant HR0011-10-C-0073)Alfred P. Sloan Foundation (Research Fellowship)United States. Defense Advanced Research Projects Agency (Young Faculty Award

The Shape of the Renormalized Trajectory in the Two-dimensional O(N) Non-linear Sigma Model

The renormalized trajectory in the multi-dimensional coupling parameter space of the two-dimensional O(3) non-linear sigma model is determined numerically under \linebreak $\delta$-function block spin transformations using two different Monte Carlo renormalization group techniques. The renormalized trajectory is compared with the straight line of the fixed point trajectory (fixed point action) which leaves the asymptotically free ultraviolet fixed point of the critical surface in the orthogonal direction. Our results show that the renormalized trajectory breaks away from the fixed point trajectory in a range of the correlation length around $\xi \approx 3$-$7$, flowing into the high temperature fixed point at $\xi=0$. The analytic large $N$ calculation of the renormalized trajectory is also presented in the coupling parameter space of the most general bilinear Hamiltonians. The renormalized trajectory in the large $N$ approximation exhibits a similar shape as in the $N=3$ case, with the sharp break occurring at a smaller correlation length of $\xi \approx 2$-$3$.Comment: 9 pages, compressed and uuencoded postscript file (compressed file is 434 Kbytes.) A reference is added with a minor modification of the text and fig.3