FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations

In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n$n$ independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented.     

Detecting symmetries of rational plane and space curves

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.     

Analysis of heregulin symmetry by weighted evolutionary tracing

Heregulins are members of the protein family of EGF-like growthand differentiation factors. The primary cell-surface targetsof heregulins are heterodimers of the EGF-receptor homolog HER2with either HER3 or HER4. We used a weighted evolutionary traceanalysis to identify structural features that distinguish theEGF-like domain (hrg) of heregulins from other members of theEGF family. In this analysis, each amino acid sequence is weightedaccording to its uniqueness and the variability in each positionis assigned by an amino acid substitution matrix. Conservedresidues in heregulin that are variable in other EGF-like domainsare considered possible specificity-conferring residues. Thisanalysis identifies two clusters of residues at the foot ofthe boot-shaped hrg domain. The residues in one cluster arerecruited from the N-terminus; those in the other are from the-loop region and show a weak sequence similarity to the N-terminalresidues at the opposite side of the boot. The remaining residueswith high conservation scores distribute themselves into thesetwo distinct surfaces on hrg. This pseudo-twofold symmetry andthe presence of two distinct interfaces may reflect the preferenceof hrg for heterodimeric versus homodimeric HER complexes.     

Dipole moments of copolymers in relation to the distribution of polar monomer unit sequences

A method for evaluating the dipole moments of copolymers in relation to the distribution of polar unit sequences is proposed. Experimental data on the synthesis and dipole moments of styrene-p-chlorostyrene and styrene-p-methoxystyrene copolymers were used to calculate the effective dipole moment of a polar unit in a copolymer. For this purpose, we assumed the following effective moment values for the corresponding polar unit sequences along the copolymer chains: when both of the nearest neighbours are polar, μ_PP, when one of the nearest neighbours is polar and the other is non-polar, μ_PS, and when both of the nearest neighbours are non-polar, μ_SS. The relative magnitudes of μ_SS, μ_PS and μ_PP calculated for both copolymers reflect the influence of dipole-dipole interactions on the effective dipole moment of a polar unit in a copolymer molecule.     

M序列的对称性变换及对称群

n级M序列是由n级非线性回馈移位寄存器产生的周期为2n的二元序列,也称为de Bruijn序列。该文将讨论M序列的对称性变换,提出了一类新的对称性变换——"与/或"逻辑对称性变换,并研究了基于对称性变换的对称群。     

离子型磁性液体中纳米微粒间相互作用研究

用磁偶板子硬球模型对磁性液体中磁性纳米微粒间相互作用进行了研究。分析表明,微粒间存在着静电排斥、静磁吸引和VanderWaals吸引三种相互作用,无外场时静磁作用非常微弱,可以忽略,静电排斥力作用距离远,起抗团聚作用,Vander Waals作用距离短,是引起团聚主要原因。为确保磁性液体稳定性,制备的离子型磁性液体体积分数应小于5％。     

框架的行置换和列置换——处理复杂系统的新思维系列之四

本系列论文基于《多边矩阵理论》,由东方整体性思维所启迪,试图提供并完善一套从整体到局部处理复杂系统多指标问题、非均匀性问题、非线性问题的强有力的数学工具,并对其进行严格的理论推导和证明。本文利用群论的相关概念和结论,将其和框架的剖分运算相结合,引入了导出框架和对称框架的定义,证明了框架的行置换等价于列置换的结论,从而以后可以统一地处理指标间的置换问题。     

基于混合模型的舰船腐蚀相关静态电、磁场

为研究舰船水下腐蚀相关静态电场、磁场的分布特征,提出了以舰船水下标量电位为线索,对舰船进行混合建模,即将舰船等效为水平电流线和离散三分量电偶极子的组合,通过测量舰船下方某平面上的标量电位分布,对混合模型中的未知结构参量进行拟合,以建好的混合模型为基础对舰船水下腐蚀相关静态电、磁场进行研究.在实验室中模拟海洋环境,利用1∶100船模对建模方法进行了验证,并以建好的混合模型为基础,计算了船模水下腐蚀相关静态电、磁场的分布特征.实验结果表明混合模型结构简洁、实测数据量小、建模计算量小、模型稳定,比较适合用于舰船腐蚀相关静态电、磁场分布特点的研究.     

基于磁梯度张量与Levenberg-Marquardt优化的磁矩计算方法

针对目前磁矩计算方法中出现的对磁性目标出厂参数精度要求较高且计算繁琐、受地磁场影响较大、反演精度较差等问题,提出了一种基于磁梯度张量概念与Levenberg-Marquardt优化算法结合的磁矩计算方法。该方法采用磁梯度张量概念消除了地磁背景场的干扰,使用Levenberg-Marquardt算法优化解算过程,消除了反演过程中出现的奇异点。对新方法进行地磁背景场、测量系统位置、环境噪声等影响因素的验证仿真,并实验验证该方法的可操作性。仿真与实验证明,相比于以往方法,新方法的磁矩反演精度有大幅度的提高,抗噪声能力更强,具有更高的鲁棒性。     

特征线坐标系下柱对称超音速流膨胀波计算方法研究

在静压圆盘止推气体轴承柱对称超音速流膨胀波系的计算中,借鉴Liepmann的特征线坐标系与自然坐标系变换理论和Liepmann引入两个黎曼不变量的方法,使得边界条件的处理更为直观和简便。采用特征线坐标系下的特征线法,推导计算该膨胀波系所用到的基本计算公式,将其应用于 P-M 流动并与 P-M 流动理论解进行比较。结果表明,数值计算结果与P-M流动理论解吻合,该计算方法可行。