On the local convergence of adjoint Broyden methods

The numerical solution of nonlinear equation systems is often achieved by so-called quasi-Newton methods. They preserve the rapid local convergence of Newton’s method at a significantly reduced cost per step by successively approximating the system Jacobian though low-rank updates. We analyze two variants of the recently proposed adjoint Broyden update, which for the first time combines the classical least change property with heredity on affine systems. However, the new update does require, the evaluation of so-called adjoint vectors, namely products of the transposed Jacobian with certain dual direction vectors. The resulting quasi-Newton method is linear contravariant in the sense of Deuflhard (Newton methods for nonlinear equations. Springer, Heidelberg, 2006) and it is shown here to be locally and q-superlinearly convergent. Our numerical results on a range of test problems demonstrate that the new method usually outperforms Newton’s and Broyden’s method in terms of runtime and iterations count, respectively. Partially supported by the DFG Research Center Matheon “Mathematics for Key Technologies”, Berlin and the DFG grant WA 1607/2-1.     

Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

We study semi-linear elliptic PDEs with polynomial non-linearity in bounded domains and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. We consider several examples and estimate their solution by using the Monte Carlo method.     

Msx2 mediates the inhibitory action of TNF-�� on osteoblast differentiation

TNF-α, a proinflammatory cytokine, inhibits osteoblast differentiation under diverse inflammatory conditions; however, the underlying mechanisms in terms of the TNF-α signaling pathway remain unclear. In this study, we examined the role of Msx2 in TNF-α-mediated inhibition of alkaline phosphatase (ALP) expression and the signaling pathways involved. TNF-α down-regulated ALP expression induced by bone morphogenetic protein 2 (BMP2) in C2C12 and Runx2^-/- calvarial cells. Over-expression of Msx2 suppressed BMP2-induced ALP expression. Furthermore, TNF-α induced Msx2 expression, and the knockdown of Msx2 by small interfering RNAs rescued ALP expression, which was inhibited by TNF-α. TNF-α activated the NF-κB and the JNK pathways. Inhibition of NF-κB or JNK activation reduced the inhibitory effect of TNF-α on ALP expression, whereas TNF-α-induced Msx2 expression was only suppressed by the inhibition of the NF-κB pathway. Taken together, these results indicate that Msx2 mediates the inhibitory action of TNF-α on BMP2-regulated osteoblast differentiation and that the TNF-α-activated NF-κB pathway is responsible for Msx2 induction.     

Stability analysis for fractional‐order partial differential equations by means of space spectral time Adams‐Bashforth Moulton method

In this article, a new numerical scheme space Spectral time Fractional Adam Bashforth Moulton method for the solution of fractional partial differential equations is offered. The proposed method is obtained by modifying, in a suitable way; the spectral technique and the method of lines. The attention is focused on the stability properties and hence an elegant stability analysis for the current approach is also provided. Finally, two examples are presented to illustrate the effectiveness of the reported method. Obtained results confirm the convergence and spectral accuracy of the proposed method in both space and time. In addition, a comparison with the existing studies is also made as a limiting case of the considered problem at the end and found in good agreement.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 19–29, 2018     

On derivatives of hypergeometric functions and classical polynomials with respect to parameters

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial.     

Synthetic Retinoids: Structure–Activity Relationships

Retinoid signalling pathways are involved in numerous processes in cells, particularly those mediating differentiation and apoptosis. The endogenous ligands that bind to the retinoid receptors, namely all‐trans‐retinoic acid (ATRA) and 9‐cis‐retinoic acid, are prone to double‐bond isomerisation and to oxidation by metabolic enzymes, which can have significant and deleterious effects on their activities and selectivities. Many of these problems can be overcome through the use of synthetic retinoids, which are often much more stable, as well as being more active. Modification of their molecular structures can result in retinoids that act as antagonists, rather than agonists, or exhibit a large degree of selectivity for particular retinoid‐receptor isotypes. Several such selective retinoids are likely to be of value as pharmaceutical agents with reduced toxicities, particularly in cancer therapy, as reagents for controlling cell differentiation, and as tools for elucidating the precise roles that specific retinoid signalling pathways play within cells.     

Monoids and decay

We propose some formalization of the concept of critical decay number and describe the class of models with this number at most 4 (i.e., every object is decomposable which consists of four or more elements).     

单位球上加权 Bergman-Nevanlinna空间到 Blo ch-型空间上乘法,复合,微分算子的乘积

文章用径向导数定义了 H(B)空间上的微分算子,从而研究了单位球上加权Bergman-Nevanlinna 空间到 Bloch-型空间上乘法,复合,微分算子的乘积,给出了这类乘积有界和紧的充要条件。     

也谈正则动量算符之争

讨论了平直空间曲线坐标系中质点运动的两种不同的量子化假设，比较了它们在物理思想和实际操作上的区别和优劣。     

牡丹草地下芽的形态分化与生长特征研究

采用解剖镜、扫描电镜技术,对长白山早春类短命植物牡丹草地下芽形态分化进行了观察,并采用游标卡尺测定各阶段芽体大小,研究地下芽的生长发育节律。发现牡丹草地上生活周期结束后,地下贮藏器官需要休眠2个月左右,于7月中下旬,地下芽开始出现形态分化和发育。其进程可分为萌芽期、花序原基分化期、花序形成期和花器官分化期4个时期,历经2个月左右,最终以发育成熟的花序进行越冬休眠。连续测定和相关分析表明,在整个夏眠期,牡丹草地下芽的茎高、芽长、芽径均随时间指数生长,茎高和芽长、芽径之间以及芽长与芽径之间呈幂函数规律异速生长,且相关系数均达到极显著水平。 牡丹草块茎需短暂休眠后完成地下芽的形态建成,地下芽茎高、芽长和芽径的增长具有不同步性和阶段性优先伸长的异速生长特性。