Asymptotics for Some Proximal-like Method Involving Inertia and Memory Aspects

Given a Hilbert space H and a closed convex function Φ:\(H\rightarrow{\mathbb{R}} \cup \{+\infty\}\), we consider the inertial proximal algorithm

$ x_{n+1}-x_n-\alpha_n(x_n-x_{n-1})+\beta_n\partial\Phi(x_{n+1})\ni 0, \qquad \qquad \qquad \qquad({\mathcal{A}}) $

where (α _n ) and (β _n ) are nonnegative sequences. The notation \(\partial \Phi\) stands for the subdifferential of Φ in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to \(\inf\Phi\) as n→?+?∞. When Φ has a unique minimum, the question of the convergence of (x _n ) is solved. In the case of multiple minima, it is proved that if \(\left(\prod_{k=1}^n \alpha_k\right)\not \in l^1\) and if a suitable geometric condition on the set argmin Φ is fulfilled, then non stationary sequences of (A) cannot converge.

     

Boundary closures for fourth-order energy stable weighted essentially non-oscillatory finite-difference schemes

A general strategy was presented in 2009 by Yamaleev and Carpenter for constructing energy stable weighted essentially non-oscillatory (ESWENO) finite-difference schemes on periodic domains. ESWENO schemes up to eighth order were developed that are stable in the energy norm for systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain, wherever possible, the WENO stencil biasing properties and satisfy the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L₂ norm. Second-order and third-order boundary closures are developed that are stable in diagonal and block norms, respectively, and achieve third- and fourth-order global accuracy for hyperbolic systems. A novel set of nonuniform flux interpolation points is necessary near the boundaries to simultaneously achieve (1) accuracy, (2) the SBP convention, and (3) WENO stencil biasing mechanics.     

Dipole‐like backscatter stimulated Raman scattering for in vivo imaging

Stimulated Raman scattering (SRS) scanning microscopy has the potential to enable label‐free in vivo imaging for research and clinical medicine. Volume SRS from focus occurs in the forward scattered direction. Therefore, multiple scattering events are required to direct the light out of the tissue, reducing imaging depth and resolution. Here, a method called Stokes interference SRS (SISRS) is introduced that operates by the addition to the standard pump and stimulated emission probe beams a third beam called the donut beam. The donut is close in wavelength to the probe beam and, after passage through a π phase plate, forms an annular beam in the focal plane with bright nodes above and below focus. The donut beats with the probe beam, and when they destructively interfere with each other, the microscope's 3‐D stimulated emission focal spot is reduced to subwavelength dimensions. A subwavelength focal volume emits a dipole pattern of SRS with forward and backscatter lobes, enabling high‐resolution single‐backscatter imaging from deep within tissues. The reduction of the focal volume also increases the resolution of the scanning image creating imaging beyond the diffraction limit. SISRS imaging may provide in vivo label‐free Raman images comparable with that achieved in stained in vitro tissues in all planes of section. Copyright © 2014 John Wiley & Sons, Ltd.     

Effect of headgroup on the supramolecular morphology of diacetylenic aldonamides

This report describes the initial characterization of supramolecular morphology of diacetylenic aldonamides where the headgroup has been systematically varied from hexose, pentose, to tetrose. The electron-dense diacetylenes allow direct electron microscopic imaging of the assembly morphology without the aid of staining reagents. The electron micrographs reveal a dramatic change in supramolecular morphology upon the simple changing the headgroup from D-galactose to L-arabinose. Possible reasons for the change in morphology are discussed.