A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.     

On the Relative Twist Formula of l-adic Sheaves

We propose a conjecture on the relative twist formula of l-adic sheaves, which can be viewed as a generalization of Kato—Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.     

分析化学教材中关于离子活度系数计算公式的讨论

在处理电解质溶液的化学平衡理论中,有关离子的活度和活度系数的计算或估算都有比较严谨的理论公式或经验公式,而国内分析化学教材中关于经验公式的表述略显随意。以0.050 mol·L^-1AlCl₃水溶液中Cl^-和Al³⁺的活度计算为例,结合国内外分析化学教材中关于活度系数计算的经验公式进行讨论。由计算结果可知,关于活度系数计算的经验公式需给出明确的使用条件。     

一种计算液接电势的新方法*

依据液接电势达稳态时通过边界层的净电荷为零这一思想,提出了一种计算液接电势的新方法。对于3类液接电势的计算,特别是Lewis-Sargent公式与Henderson方程的推导,思路清晰易懂,避免了过往液接电势计算方法存在的一些问题。本文所提出的方法能够为液接电势相关内容的教学提供借鉴。     

轨形Gromov-Witten不变量沿光滑点的加权涨开公式

本文考虑,当一个紧辛轨形群胚(X,ω)沿着光滑点作加权涨开时,它的形如<α_1,…,α_m,[pt]>_(g,A)~X的轨形Gromov-Witten不变量的变化公式,其中[pt]∈H_(dR)~(2n)(X)是生成元,dimX=2n.我们证明了对于非零A∈H_2(|X|,Z),<α_1,…,α_m,[pt]>_(g,A)~X={_(g_1,pl(A)-e’)~xdimX=4,g≥0,∑((-1)g_1·2)/(2g_1+2)!_(g_2,pl(A)-e’)~xdimX=6,g≥0,_(g_1,pl(A)-e’)~xdimX≥8,g=0其中x是X沿一光滑点的权α=(α_1,…,α_n)的加权涨开,且α_1≥α_i,2≤i≤n.     

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

We present a second‐order ensemble method based on a blended three‐step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second‐order ensemble method that combines the two‐step BDF timestepping scheme and a special explicit second‐order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 34–61, 2017     

三项指数和四次均值的精确计算公式(2)

本文进一步深入研究了三项指数和四次均值的计算问题.运用指数和的相关性质并结合求解同余方程组的方法与技巧,利用两种不同的方法获得了两个精确的均值计算公式,揭示了三项指数和的计算与同余方程组解的个数之间的本质联系,推广了已有的结果.     

二项指数和四次混合均值的计算

本文研究了二项指数和四次均值的计算问题.利用初等数论及代数数论的方法获得了一个精确的计算公式以及一个转换公式,推广了已有的结果,揭示了均值计算与同余方程组的本质联系.     

基于三次Bézier基函数插值的GM(1,1)模型背景值优化研究

本文研究了传统灰色GM(1,1)模型存在模型精度不高的问题.利用带形状参数的三次Bézier基函数,给出插值函数的表达式,并结合复化梯形公式,给定误差限的方法,获得了比传统灰色GM(1,1)模型更高精度的结果.推广了传统灰色GM(1,1)预测模型的结果.