Numerical approach to chaotic pattern formation in diffusive predator–prey system with Caputo fractional operator

This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices.     

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.     

A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations

The $p$-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallel-in-time preconditioned Krylov subspace solvers (see McDonald et al. [36] and Lin and Ng [32]). However, when the BDF$p$ (2 ≤ $p$ ≤ 6) method is used to solve time-dependent PDEs, the generalization of these studies is not straightforward as $p$-step BDF is not selfstarting for $p$ ≥ 2. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, and show that it results into an all-at-once discretized system that is a low-rank perturbation of a block triangular Toeplitz system. We first give an estimation of the condition number of the all-at-once systems and then, capitalizing on previous work, we propose two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. An efficient implementation of these BC preconditioners is also presented, including the fast computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.     

${[\ell_p]}_{e.r}$ Euler-Riesz Difference Sequence Spaces

Başar and Braha [1], introduced the sequence spaces $\breve{\ell}_\infty$, $\breve{c}$ and $\breve{c}_0$ of Euler-Cesáro bounded, convergent and null difference sequences and studied their some properties. Then, in [2], we introduced the sequence spaces ${[\ell_\infty]}_{e.r}, {[c]}_{e.r}$ and ${[c_0]}_{e.r}$ of Euler-Riesz bounded, convergent and null difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. The main purpose of this study is to introduce the sequence space ${[\ell_p]}_{e.r}$ of Euler-Riesz $p-$absolutely convergent series, where $1 \leq p <\infty$, difference sequences by using the composition of the Euler mean $E_1$ and Riesz mean $R_q$ with backward difference operator $\Delta$. Furthermore, the inclusion $\ell_p\subset{[\ell_p]}_{e.r}$ hold, the basis of the sequence space ${[\ell_p]}_{e.r}$ is constructed and $\alpha-$, $\beta-$ and $\gamma-$duals of the space are determined. Finally, the classes of matrix transformations from the ${[\ell_p]}_{e.r}$ Euler-Riesz difference sequence space to the spaces $\ell_\infty, c$ and $c_0$ are characterized. We devote the final section of the paper to examine some geometric properties of the space ${[\ell_p]}_{e.r}$.     

拉曼光谱和色差法在丝质文物色彩损伤评估中的比较

丝质文物是博物馆内的重要藏品之一,具有极高的文化、艺术、历史价值。作为一种性质不稳定的蛋白质有机构成材料,丝质文物极易受到光学辐射而发生泛黄等色彩损伤,尤其是在Light Emitting Diode(LED)得到广泛应用的博物馆光环境中。如何针对馆藏丝质文物色彩损伤进行科学评估是本研究要解决的主要问题。色差评估法是分析博物馆照明对文物色彩损伤的有效手段,但是存在相应的局限性,无法对色彩损伤的诱导期进行评估。由于文物发生色彩损伤的根本原因在于材料内部分子结构发生光化学反应,理论上说,从微观分子层面研究的拉曼光谱法能够更加科学地评估丝质文物色彩损伤。本研究将拉曼光谱引入博物馆照明领域,对比色差评估结果验证拉曼光谱色彩损伤评估的可行性和科学性。通过构成四基色LED的450, 510, 583和650 nm四种窄带光对丝质样品开展长周期照射实验,分别以CIE L*a*b*色差和拉曼光谱作为评估指标,计算得到基于两种评估方法的不同窄带光对丝质文物的色彩相对损伤系数,分别为450 nm∶510 nm∶583 nm∶650 nm=1.00∶0.63∶0.48∶0.32和450 nm∶510 nm∶583 nm∶650 nm=1.00∶0.69∶0.47∶0.27。结果一方面表明,两种方法得到的四种窄带光对丝质文物的色彩损伤趋势是一致的,即450 nm>510 nm>583 nm>650 nm,波长越短对丝质文物的色彩损伤程度越大,说明拉曼光谱是一种能够量化评估丝质文物色彩损伤的可行方法;另一方面表明,基于拉曼参数计算的色彩相对损伤系数的比例差异更大。丝质样品的老化过程中存在诱导期。色差法难以分析丝质样品诱导期的色彩变化,而拉曼光谱分析则可灵敏地检测出相应的分子结构变化,包括诱导期。因此,拉曼光谱法能够对丝质文物的色彩损伤进行更加科学地评估。同时,研究得到的构成LED光谱的四种窄带光对丝质文物色彩相对损伤系数,可以为这类光源在丝质文物照明中的损伤度评估和博物馆准入评估提供依据。     

Immunocytochemical Localization of XRCC1 and γH2AX Foci Induced by Tightly Focused Femtosecond Laser Radiation in Cultured Human Cells

To assess the prospects for using intense femtosecond laser radiation in biomedicine, it is necessary to understand the mechanisms of its action on biological macromolecules, especially on the informational macromolecule—DNA. The aim of this work was to study the immunocytochemical localization of DNA repair protein foci (XRCC1 and γH2AX) induced by tightly focused femtosecond laser radiation in human cancer A549 cells. The results showed that no XRCC1 or γH2AX foci tracks were observed 30 min after cell irradiation with femtosecond pulses of 10¹¹ W∙cm⁻² peak power density. An increase in the pulse power density to 2 × 10¹¹ W∙cm⁻² led to the formation of linear tracks consisting both of XRCC1 and γH2AX protein foci localized in the places where the laser beam passed through the cell nuclei. A further increase in the pulse power density to 4 × 10¹¹ W∙cm⁻² led to the appearance of nuclei with total immunocytochemical staining for XRCC1 and γH2AX on the path of the laser beam. Thus, femtosecond laser radiation can be considered as a tool for local ionization of biological material, and this ionization will lead to similar effects obtained using ionizing radiation.     

Far-infrared free-electron lasers and their applications

Free-electron lasers based on radio-frequency linear accelerators provide an important source of far-infrared radiation which allow exciting experiments that cannot be performed in any other way. Facilities such as FELIX (Nieuwegein, The Netherlands), JFEL (Newport News VA, USA), FELBE (Dresden, Germany), CLIO (Paris, France) and others provide mid- and far-infrared output in picosecond pulses with micro-joules of energy. They give continuous, wide tuning in far-IR for resonant pumping of discrete transitions (with simultaneous coverage of mid-IR) from around 3 to 250 μm wavelength. This enables time-resolved spectroscopy, non-linear optics and spectroscopy of weak absorptions. They have been applied to a wide variety of problems in condensed matter physics, physical chemistry and biophysics. We review the physics applications of these sources.     

Plasmonic planar lens based on stack of metal–insulator–metal waveguides with height tuning

We introduce a technique capable of focusing electromagnetic (EM) waves through plasmonic nanoslits symmetrically arranged along the indented semi-circular surface in silver background. The EM transports through the tuning slits in the form of surface plasmon polaritons (SPPs), and gets the required phase retardations to focus at the focal plane. Due to the subwavelength nature of planar metallic lens, we present the rigorous electromagnetic analysis by using two dimensional (2D) finite difference time domain (FDTD) method. These height-modulated slits with uniform width are demonstrated to have unique advantages in beam manipulation. In combination with previous studies, it is expected that our structure with small number of slits could lead to realization of optimum designs of plasmonic nano-lens.     

High-pressure and high-temperature generation using diamond/sic composite anvils prepared with hot isostatic pressing

Using a hot isostatic pressing (HIP) technique, we synthesized diamond/SiC composites from diamond and Si powders. At an HIP condition of 1450 °C and 100 MPa, a pressure much lower than that of the diamond stability field, diamond powders react with molten Si to form well-sintered diamond/SiC composites. Cubes of the composites with 15 mm edge length were thereby fabricated, and an application to the second stage anvils in a Kawai-type high-pressure apparatus was attempted. A hybrid anvils system using four cubes of the composites and four of the conventional WC was introduced and heating experiments up to 1600 °C became possible. Because the diamond/SiC composites are transparent to X-rays, the present system is applicable not only to diffraction studies but also to radiographic studies that need a larger window for an X-ray image.     

Study of partial melting at high-pressure using in situ X-ray diffraction

The high-pressure melting behavior of different iron alloys was investigated using the classical synchrotron-based in situ X-ray diffraction techniques. As they offer specific advantages and disadvantages, both energy-dispersive (EDX) and angle-dispersive (ADX) X-ray diffraction methods were performed at the BL04B1 beamline of SPring8 (Japan) and at the ID27-30 beamline of the ESRF (France), respectively. High-pressure vessels and pressure ranges investigated include the Paris–Edinburgh press from 2 to 17 GPa, the SPEED-1500 multi-anvil press from 10 to 27 GPa, and the laser-heated diamond anvil cell from 15 to 60 GPa. The onset of melting (at the solidus or eutectic temperature) can be easily detected using EDX because the grains start to rotate relative to the X-ray beam, which provokes rapid and drastic changes with time of the peak growth rate. Then, the degree of melting can be determined, using both EDX and ADX, from the intensity of diffuse X-ray scattering characteristic of the liquid phase. This diffuse contribution can be easily differentiated from the Compton diffusion of the pressure medium because they have different shapes in the diffraction patterns. Information about the composition and/or about the structure of the liquid phase can then be extracted from the shape of the diffuse X-ray scattering.