Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

基于遗传算法的非线性映射预测硝基苯化合物的生物毒性

摘要: 非线性映射是一种降维处理技术,通过Sammon算法可以将多维分析问题转化为二维问题分析。应用该法的关键在于寻求二维空间的最佳投影点,这可以转化为一个复杂的非线性优化问题来进行解决。遗传算法是解决各类函数优化和非线性映射问题的一种有效算法。以硝基苯化合物的生物毒性的模式识别为例,编制基于误差函数的目标函数和遗传算法约束参数,应用基于MATLAB的遗传算法工具箱进行优化求解。结果表明:该工具箱在求解此类非线性优化问题上是有效的, 基于遗传算法的非线性映射的预测结果比其它方法要好些。     

Third-order nonlinear optical studies of new anthraquinone derivatives

Four new donor-acceptor-donor anthraquinone derivatives were synthesized and characterized by UV, IR, ¹H NMR and elemental analysis. Their off-resonant third-order nonlinear optical properties were measured using femtosecond laser and degenerate four-wave mixing technique. The third-order nonlinear optical susceptibilities χ⁽³⁾ were 3.36-3.76 × 10⁻¹³ esu. The nonlinear refractive indexes n₂ were 6.19-6.91 × 10⁻¹² esu. The second-order hyperpolarizabilities γ of the molecules were 3.36-3.76 × 10⁻³¹ esu. The response times were 101-115 fs. The results show that these compounds have potential nonlinear optical applications.     

Adsorption of nonpolar benzene derivatives on single-walled carbon nanotubes

The adsorption of benzene, toluene, and chlorobenzene on single-walled carbon nanotubes (SWCNTs) with and without acid oxidation was conducted to investigate the influences of derivative groups on benzene rings and functional groups from SWCNTs on adsorption by SWCNTs. The SWCNTs of high purity were chosen and moderate acid oxidation was performed so that the surface physical properties remained unchanged after acid oxidation and the influences of acid oxidation on adsorption were only contributed from the modification of the surface chemistry of SWCNTs. The oxygen-containing surface groups introduced by acid oxidation obstructed the interactions between functional groups of nonpolar benzene derivatives and C-rings of SWCNTs significantly. The dispersive interaction between the partially positive H⁺ of the methyl group and the oxygen-containing surface groups slightly increased the adsorption of toluene on oxidized SWCNTs at high solution pH. The thermodynamic of adsorption was also studied at different temperatures.     

A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law

The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV-related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV-TB coinfected model in the Atangana-Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers-Ulam (HU) stability and numerical simulations with assumption of specific parameters.     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Fast Evaluation of Linear Combinations of Caputo Fractional Derivatives and Its Applications to Multi-Term Time-Fractional Sub-Diffusion Equations

In the present work, linear combinations of Caputo fractional derivatives are fast evaluated based on the efficient sum-of-exponentials (SOE) approximation for kernels in Caputo fractional derivatives with an absolute error $\epsilon,$ which is a further work of the existing results in [13] (Commun. Comput. Phys., 21 (2017), pp. 650-678) and [16] (Commun. Comput. Phys., 22 (2017), pp. 1028-1048). Both the storage needs and computational amount are significantly reduced compared with the direct algorithm. Applications of the proposed fast algorithm are illustrated by solving a second-order multi-term time-fractional sub-diffusion problem. The unconditional stability and convergence of the fast difference scheme are proved. The CPU time is largely reduced while the accuracy is kept, especially for the cases of large temporal level, which is displayed by numerical experiments.     

Uniqueness of meromorphic functions concerning sharing two small functions with their derivatives

In this paper, we study the uniqueness of meromorphic functions that share two small functions with their derivatives. We prove the following result: Let $f$ be a nonconstant meromorphic function such that $\mathop {\overline{\lim}}\limits_{r\to\infty} \frac{\bar{N}(r,f)}{T(r,f)}<\frac{3}{128}$, and let $a$, $b$ be two distinct small functions of $f$ with $a\not\equiv\infty$ and $b\not\equiv\infty$. If $f$ and $f"$ share $a$ and $b$ IM, then $f\equiv f"$.     

Mesophases in a Gel from Hydroxypropyl Cellulose/Polyacrylamide

The objective of the present work is to analyze the different mesophases observed in a gel synthesized from hydroxypropyl cellulose (HPC) and polyacrylamide (PAAm) and its phase behavior in water. Hydroxypropyl cellulose is a material derived from cellulose it is non toxic and degradable. HPC is widely used in pharmaceutics, food additives, stabilizer, thickener, etc., HPC is approved by the FDA for use in the food industry. HPC has the characteristic that it forms liquid crystals, depending on the solvent and on the concentration of the polymer. In this work a lower critical solution temperature (LCST) and an upper critical solution temperature (UCST) were observed, the resulting gel showed anisotropic, nematic and cholesteric phases in water, the phase formed depended on the concentration of the polymer.     

Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.