Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations

In this work, we extend the fractional linear multistep methods in [C. Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional integral and derivative operators in the sense that the tempered fractional derivative operator is interpreted in terms of the Hadamard finite-part integral. We develop two fast methods, Fast Method I and Fast Method II, with linear complexity to calculate the discrete convolution for the approximation of the (tempered) fractional operator. Fast Method I is based on a local approximation for the contour integral that represents the convolution weight. Fast Method II is based on a globally uniform approximation of the trapezoidal rule for the integral on the real line. Both methods are efficient, but numerical experimentation reveals that Fast Method II outperforms Fast Method I in terms of accuracy, efficiency, and coding simplicity. The memory requirement and computational cost of Fast Method II are $O(Q)$ and $O(Qn_T)$, respectively, where $n_T$ is the number of the final time steps and $Q$ is the number of quadrature points used in the trapezoidal rule. The effectiveness of the fast methods is verified through a series of numerical examples for long-time integration, including a numerical study of a fractional reaction-diffusion model

Heliotracking Device using Liquid Crystalline Elastomer Actuators

Many living organisms in nature respond to light stimulus and track the light source. Inspired by this, and to maximize the light-harvesting ability of solar cells, here, a spontaneous helio-tracking device based on the differential light-induced actuation of liquid crystalline elastomers (LCEs) is demonstrated. The synthesis of the actuator material involves a robust thiol “click” polymerization, while the addition of indocyanine green (ICG) dye imparts the sensitivity to broad-spectrum and near-infrared light. Highly reproducible thermal and photo-induced linear actuation is demonstrated. The device is based on a freely pivoting payload platform held in place by several linear LCE actuators around the 360° circumference. The side of the device, which is exposed to light, has the actuators contracting and tilting the platform toward the light source. As the light source (e.g., the Sun) is moving around the device, the platform tilt followed, always exposing the payload face to the light; in the dark, the device recovers its neutral position

Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

Automatic quantitative analysis of experimental primary and secondary retinal neurodegeneration: implications for optic neuropathies.

Secondary neurodegeneration is thought to play an important role in the pathology of neurodegenerative disease, which potential therapies may target. However, the quantitative assessment of the degree of secondary neurodegeneration is difficult. The present study describes a novel algorithm from which estimates of primary and secondary degeneration are computed using well-established rodent models of partial optic nerve transection (pONT) and ocular hypertension (OHT). Brn3-labelled retinal ganglion cells (RGCs) were identified in whole-retinal mounts from which RGC density, nearest neighbour distances and regularity indices were determined. The spatial distribution and rate of RGC loss were assessed and the percentage of primary and secondary degeneration in each non-overlapping segment was calculated. Mean RGC number (82 592±681) and RGC density (1695±23.3 RGC/mm(2)) in naïve eyes were comparable with previous studies, with an average decline in RGC density of 71±17 and 23±5% over the time course of pONT and OHT models, respectively. Spatial analysis revealed greatest RGC loss in the superior and central retina in pONT, but significant RGC loss in the inferior retina from 3 days post model induction. In comparison, there was no significant difference between superior and inferior retina after OHT induction, and RGC loss occurred mainly along the superior/inferior axis (~30%) versus the nasal-temporal axis (~15%). Intriguingly, a significant loss of RGCs was also observed in contralateral eyes in experimental OHT. In conclusion, a novel algorithm to automatically segment Brn3a-labelled retinal whole-mounts into non-overlapping segments is described, which enables automated spatial and temporal segmentation of RGCs, revealing heterogeneity in the spatial distribution of primary and secondary degenerative processes. This method provides an attractive means to rapidly determine the efficacy of neuroprotective therapies with implications for any neurodegenerative disorder affecting the retina

Polynomial Carleson operators along monomial curves in the plane

We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem.Comment: 27 page

Role of domain walls in the abnormal photovoltaic effect in BiFeO3

Recently, the anomalous photovoltaic (PV) effect in BiFeO3 (BFO) thin films, which resulted in open circuit voltages (V-oc) considerably larger than the band gap of the material, has generated a revival of the entire field of photoferroelectrics. Here, via temperature-dependent PV studies, we prove that the bulk photovoltaic (BPV) effect, which has been studied in the past for many non-centrosymmetric materials, is at the origin of the anomalous PV effect in BFO films. Moreover, we show that irrespective of the measurement geometry, V-oc as high as 50V can be achieved by controlling the conductivity of domain walls (DW). We also show that photoconductivity of the DW is markedly higher than in the bulk of BFO

Annexins in glaucoma

Glaucoma is one of the leading causes of irreversible visual loss, which has been estimated to affect 3.5% of those over 40 years old and projected to affect a total of 112 million people by 2040. Such a dramatic increase in affected patients demonstrates the need for continual improvement in the way we diagnose and treat this condition. Annexin A5 is a 36 kDa protein that is ubiquitously expressed in humans and is studied as an indicator of apoptosis in several fields. This molecule has a high calcium-dependent affinity for phosphatidylserine, a cell membrane phospholipid externalized to the outer cell membrane in early apoptosis. The DARC (Detection of Apoptosing Retinal Cells) project uses fluorescently-labelled annexin A5 to assess glaucomatous degeneration, the inherent process of which is the apoptosis of retinal ganglion cells. Furthermore, this project has conducted investigation of the retinal apoptosis in the neurodegenerative conditions of the eye and brain. In this present study, we summarized the use of annexin A5 as a marker of apoptosis in the eye. We also relayed the progress of the DARC project, developing real-time imaging of retinal ganglion cell apoptosis in vivo from the experimental models of disease and identifying mechanisms underlying neurodegeneration and its treatments, which has been applied to the first human clinical trials. DARC has potential as a biomarker in neurodegeneration, especially in the research of novel treatments, and could be a useful tool for the diagnosis and monitoring of glaucoma