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We consider the energy supercritical defocusing nonlinear Schrödinger equation i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension d ≥ 5 . In a suitable range of energy supercritical parameters ( d ,  p ), we prove the existence of C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism . Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

In the principle of lens imaging, when we project a three-dimensional object onto a photosensitive element through a convex lens, the point intersecting the focal plane can show a clear image of the photosensitive element, and the object point far away from the focal plane presents a fuzzy image point. The imaging position is considered to be clear within the limited size of the front and back of the focal plane. Otherwise, the image is considered to be fuzzy. In microscopic scenes, an electron microscope is usually used as the shooting equipment, which can basically eliminate the factors of defocus between the lens and the object. Most of the blur is caused by the shallow depth of field of the microscope, which makes the image defocused. Based on this, this paper analyzes the causes of defocusing in a video microscope and finds out that the shallow depth of field is the main reason, so we choose the corresponding deblurring method: the multi-focus image fusion method. We proposed a new multi-focus image fusion method based on sparse representation (DWT-SR). The operation burden is reduced by decomposing multiple frequency bands, and multi-channel operation is carried out by GPU parallel operation. The running time of the algorithm is further reduced. The results indicate that the DWT-SR algorithm introduced in this paper is higher in contrast and has much more details. It also solves the problem that dictionary training sparse approximation takes a long time.

We consider the defocusing Calogero--Moser derivative nonlinear Schr{\"o}dinger equation\\begin{align*}i \\partial_{t} u+\\partial_{x}^2 u-2\\Pi D\\left(|u|^{2}\\right)u=0, \\quad (t,x ) \\in \\mathbb{R} \\times \\mathbb{R}\\end{align*}posed on \\(E := \\left\\{u \\in L^{\\infty}(\\mathbb{R}): u' \\in L^{2}(\\mathbb{R}), u'' \\in L^{2}(\\mathbb{R}), |u|^{2}-1 \\in L^{2}(\\mathbb{R})\\right\\}\\). We prove the global well-posedness of this equation in \\(E\\). Moreover, we give an explicit formula for the chiral solution to this equation.

Consider the defocusing quintic nonlinear Schr\"{o}dinger equation on ${\\bf R}^3$ with initial data in the energy space. This problem is ``energy-critical'' in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.

Utilizing the Riemann-Hilbert (RH) approach, we shed light on the spectral structure of a defocusing shifted nonlocal NLS equation with a space-shifted parameter from which we derive its soliton solutions. The spectral structure involves the scattering data and their corresponding symmetry relations. Firstly, by performing spectral analysis of the corresponding Lax pair, we explore in detail the spectral structure of the defocusing shifted nonlocal NLS equation. It is shown that the zeros of the RH problem of the defocusing shifted nonlocal NLS equation do not allow for purely imaginary ones, which is rather different from its focusing counterpart. Secondly, based on the revealed spectral structure, the even-order soliton solutions of the defocusing shifted nonlocal NLS equation are rigorously obtained. Thirdly, the dynamical properties underlying the obtained soliton solutions are analyzed and then graphically illustrated by highlighting the role that the space-shifted parameter plays.

The direct laser metal deposition (DLMD) is an additive manufacturing technology, based on laser cladding, which focuses mainly on 3D manufacturing applications. DLMD allows the production of thin-walled components by overlaying single-track depositions. Several issues can affect the deposition process and compromise the flatness of the surface on which subsequent tracks will be deposited. This work focused on deposition troubles simulated by means of a designed variation of the standoff distance and the laser defocusing distance. The effects of these two important process parameters on the deposition process were investigated. The experimental tests were performed by depositing a nickel-based superalloy powder on AISI 304 stainless steel plates through a coaxial nozzle. The work was carried out using an ytterbium fiber laser source and a deposition head equipped with an advanced and innovative motorized optics system. This allows the decoupled variation of the laser defocusing distance and consequently the laser spot size on the substrate surface with respect to the standoff distance. Results showed an influence of standoff distance and laser defocusing distance on the geometrical characteristics of the clad, such as clad width, clad height, penetration depth, and dilution. An experimental setup consisting of a light coaxial to the powder flow and a laterally positioned camera was designed to investigate the spatial powder distribution. Moreover, an analytical model for the powder distribution and clad width were proposed and validated. The analysis of variance (ANOVA) with a general linear model was also employed to describe the results.

While the currents of the defocusing Ablowitz-Ladik chain are unambiguously defined, no explicit formulas for their densities have been obtained yet. We close this gap. As a consequence, we also complete the argument which establishes the anticipated expression for average currents, averaged over a generalized Gibbs ensemble.

Active planar optical devices that can dynamically manipulate light are highly sought after in modern optics and nanophotonics. The geometric phase derived from the photonic spin-orbit interaction provides an integrated strategy. Corresponding elements usually suffer from static functions. Here, we introduce an inhomogeneously self-organized anisotropic medium featured by photo-invertible chiral superstructure to realize geometric phase elements with continuously tunable working spectrum and light-flipped phase profile. Via preprograming the alignment of a cholesteric liquid crystal mixed with a photo-responsive chiral dopant, we demonstrate light-activated deflector, lens, Airy beam and optical vortex generators. Their polychromatic working bands are reversibly tuned in an ultra-broadband over 1000 nm covering green to telecomm region. The chirality inversion triggers facile switching of functionalities, such as beam steering, focusing/defocusing and spin-to-orbital angular momentum conversion. This work offers a platform for advanced adaptive and multifunctional flat optics with merits of high compactness, low loss and broad bandwidth. Optically reconfigurable elements are in demand for future applications. The authors report on the use of chirality-invertible cholesteric liquid crystals to actively manipulate geometric phase and create switchable planar optics elements that perform a variety of functions.

Inspired by insect compound eyes (CEs) that feature unique optical schemes for imaging, there has recently been growing interest in developing optoelectronic CE cameras with comparable size and functions. However, considering the mismatch between the complex 3D configuration of CEs and the planar nature of available imaging sensors, it is currently challenging to reach this end. Here, we report a paradigm in miniature optoelectronic integrated CE camera by manufacturing polymer CEs with 19~160 logarithmic profile ommatidia via femtosecond laser two-photon polymerization. In contrast to μ-CEs with spherical ommatidia that suffer from defocusing problems, the as-obtained μ-CEs with logarithmic ommatidia permit direct integration with a commercial CMOS detector, because the depth-of-field and focus range of all the logarithmic ommatidia are significantly increased. The optoelectronic integrated μ-CE camera enables large field-of-view imaging (90°), spatial position identification and sensitive trajectory monitoring of moving targets. Moreover, the miniature μ-CE camera can be integrated with a microfluidic chip and serves as an on-chip camera for real-time microorganisms monitoring. The insect-scale optoelectronic μ-CE camera provides a practical route for integrating well-developed planar imaging sensors with complex micro-optics elements, holding great promise for cutting-edge applications in endoscopy and robot vision. The defocusing problem has been considered the main bottleneck for developing optoelectronic μ-compound eye (CE) cameras. Here, the authors report miniature optoelectronic CE cameras with an ommatidia logarithmic-profile. The camera enables large field-of-view imaging, spatial position identification, and sensitive trajectory monitoring of moving targets.

In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small 'escape window' in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.