Weighted Hermite–Hadamard–Mercer type inequalities for convex functions

In this paper, first, we prove the weighted Hermite–Hadamard–Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right‐sides of weighted Hermite–Hadamard–Mercer type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. The results presented here would provide extensions of those given in earlier works.     

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.     

临界或超临界增长分数阶Schrodinger-Poisson方程正解的存在性

本文研究如下分数阶Schrodinger-Poisson方程{(-△)^su+Vx(u)+K(x)φu=f(u)+λ|u|^q-2ux∈R³,(-△)^tφ=K(x)u²,x∈R³其中S∈(3/4,1),t∈(0,1),f是在原点超线性无穷远次临界的连续非线性项,指数q≥2_s^*=6/3-2x.当λ>0充分小时,我们利用变分方法证明上述问题正解的存在性.本文的主要贡献是处理了超临界情形.     

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q ‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology.     

基于Ph,e下Riemann-Liouville分数阶微分方程解的存在唯一性

利用基于集合P_h,e上的一类混合单调算子不动点定理,研究了一类Riemann Liouville分数阶微分方程两点边值问题,获得了这类方程在集合P_h,e中解的存在性与唯一性,并用一组单调迭代序列逼近了该方程的唯一非平凡解.最后,利用一个实例验证了主要结论.     

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.     

Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.     

基于FT-ICR质谱仪的光解离光谱自动化谱图采集程序的开发与应用

光解离光谱在气相离子结构及动力学研究中发挥着重要的作用。这种方法非常灵活,可在实验室中利用不同种类的激光器及质谱仪的结合来开展相应的研究。但也存在一些问题,例如,自动化水平低,费时,长时间的操作容易引入人为错误,使实验结果的可信度降低。为解决此类问题,该工作开发了一套可用于光谱自动采集和分析的程序AutoMS。该程序包含：(1)AutoSpecMS用于在无需改变原仪器配套的商业化操作软件的前提下,实现多台商业激光器及一台商业高分辨FT-ICR质谱仪的集成,只需要预设参数即可自动获取action光谱;(2)AutoDataMS用于实验数据的分析,可以通过一维、二维或三维图示的方法来展示光解离质谱和光谱等。为验证改程序的可行性,针对tetraphenylpyrrin(四苯基卟啉,TPP),tetra(4-carboxypyryl)porphyrin(四对羧基苯基卟啉,TCPP)以及tetra(4-aminophenyl)porphyrin(四对氨基苯基卟啉,TCPP)三种卟啉衍生物分别开展了相应的实验。使用光解离光谱自动采集程序采集了TPP, TCPP和TAPP在210~700 nm波段的紫外-可见光解离光谱。同时获得了相关光解离碎片离子的单通道光谱,有利于用户对所研究体系的解离机理及解离动力学等方面开展深入的分析探讨。光解离质谱可观察到目标离子在特定波长激光照射下的具体解离碎片,与光解离光谱相结合,可以更好地帮助研究者理解离子在特定波段内的光解离行为。使用AutoDataMS可分别获得样品的2D及3D光解离质谱-光谱,增强了结果的可视化,使研究者更方便地获得光谱中的相关信息。对数据进行进一步分析可看出,不同的取代基对紫外可见光谱的吸收产生了差异明显的影响。此外通过AutoDataMS程序的功能扩展,还实现了对质子化TPP在500～650 nm波段内各离子的相关度分析,加深了对实验数据的理解。由此可见该方法及程序具有较好的扩展性和实用性,并可以为其他相关的工作提供很好的参考。     

The Occurrence and Biological Activity of Tormentic Acid—A Review

This review focuses on the natural sources and pharmacological activity of tormentic acid (TA; 2α,3β,19α-trihydroxyurs-2-en-28-oic acid). The current knowledge of its occurrence in various plant species and families is summarized. Biological activity (e.g., anti-inflammatory, antidiabetic, antihyperlipidemic, hepatoprotective, cardioprotective, neuroprotective, anti-cancer, anti-osteoarthritic, antinociceptive, antioxidative, anti-melanogenic, cytotoxic, antimicrobial, and antiparasitic) confirmed in in vitro and in vivo studies is compiled and described. Biochemical mechanisms affected by TA are indicated. Moreover, issues related to the biotechnological methods of production, effective eluents, and TA derivatives are presented.     

Reactive Nanoparticles Derived from Polysaccharide Phenyl Carbonates

Polysaccharide (PS) based nanoparticles (NP) are of great interest for biomedical applications. A key challenge in this regard is the functionalization of these nanomaterials. The aim of the present work was the development of reactive PS-NP that can be coupled with an amino group containing compounds under mild aqueous conditions. A series of cellulose phenyl carbonates (CPC) and xylan phenyl carbonates (XPC) with variable degrees of substitution (DS) was obtained by homogeneous synthesis. The preparation of PS-NP by self-assembling of these hydrophobic derivatives was studied comprehensively. While CPC mostly formed macroscopic aggregates, XPC formed well-defined spherical NP with diameters around 100 to 200 nm that showed a pronounced long-term stability in water against both particle aggregation as well as cleavage of phenyl carbonate moieties. Using an amino group functionalized dye it was demonstrated that the novel XPC-NP are reactive towards amines. A simple coupling procedure was established that enables direct functionalization of the reactive NP in an aqueous dispersion. Finally, it was demonstrated that dye functionalized XPC-NP are non-cytotoxic and can be employed in advanced biomedical applications.