守恒高阶各向异性交通流模型基于POD方法的降阶外推差分格式

利用Godunov流方法和特征投影分解方法,对守恒高阶各向异性交通流模型建立一种自由度很少、精度足够高的降阶外推差分算法, 并给出这种降阶外推差分算法近似解的误差估计和算法实现.最后,用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶外推差分算法的优越性.     

二维土壤溶质输运方程基于POD方法的降阶CN有限体积元外推算法

利用Crank-Nicolson(CN)有限体积元方法和特征投影分解方法建立二维土壤溶质输运方程的一种维数很低、精度足够高的降阶CN有限体积元外推算法,并给出这种外推算法的降阶CN有限体积元解的误差估计和算法的实现.最后用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶CN有限体积元外推算法的优越性.     

Sobolev方程基于POD的降阶外推差分算法

用奇异值分解和特征投影分解(proper orthogonal decomposition,简记POD)方法建立Sobolev方程的一种降阶外推有限差分算法,并给出误差估计.最后用数值例子,验证基于POD方法降阶外推有限差分算法的可行性和有效性.     

抛物方程基于POD方法的降阶外推差分算法

用奇值分解和特征投影分解(Proper Orthogonal Decomposition,简记POD)方法去建立抛物方程的一种降阶外推有限差分算法,并给出误差估计.最后用数值例子验证这种基于POD方法降阶外推有限差分算法的可行性和有效性.     

有关龙贝格求积方法的一点思考

龙贝格求积公式是《数值分析》内容中重要的数值积分方法.复化梯形公式可以通过理查森外推得到复化辛普森公式,复化辛普森公式外推可以得到复化柯特斯公式,因此,学生很自然地想:复化柯特斯公式可以继续外推,得到的会不会就是复化8阶牛顿-柯特斯公式?这里将给出具体的推导证明,从而帮助学生更好地理解龙贝格算法,更好地使用龙贝格算法.     

多重积分数值计算的梯形法余项及外推算法

研究了应用梯形法进行多重积分数值计算的余项的一般形式,为多重积分的外推算法提供了理论依据,同时提出了一种按积分变量逐维外推的数值计算方法.     

新外推完全多重网格法

使用新外推公式和高阶插值算子,为相邻细层提供好的初值,对初值使用磨光算子磨光几次后,再调用V型多重网格法求得该层数值解,构造了基于四阶紧致差分格式的新外推完全多重网格法.数值实验表明,与对比算法相比,新算法迭代次数少、计算时间短、稳健性强.     

刘徽数学思想的新认识——纪念刘徽“割圆术”1753周年

刘徽《割圆术》(公元263年)千古之谜被破解,发现使用了外推法,由此引发新的认识.将综述刘徽的极限和外推思想,并比较了国外的工作.最后指出如何用外推预报提出求解偏微分方程的高效算法.     

非饱和土壤水流方程基于特征投影分解方法的降阶CN有限元外推模型

利用Crank-Nicolson有限元方法和特征投影分解方法去建立二维非饱和土壤水流方程的一种维数很低,精度足够高的降阶CN有限元外推模型,并给出这种降阶CN有限元外推模型的降阶近似解误差估计和算法实现.最后用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶CN有限元外推模型的优越性.     

新外推完全多重网格法北大核心

使用新外推公式和高阶插值算子,为相邻细层提供好的初值,对初值使用磨光算子磨光几次后,再调用V型多重网格法求得该层数值解,构造了基于四阶紧致差分格式的新外推完全多重网格法.数值实验表明,与对比算法相比,新算法迭代次数少、计算时间短、稳健性强.     

信用传染违约Aalen加性风险模型

本文考虑了基于加性风险模型的信用风险违约预报模型,不但考虑了宏观因素和公司个体因素,并且通过引入行业因素来刻画公司间可能存在的不同于宏观因素的信用传染效应,由此克服了以往模型对违约相关性的低估.本文在参数加性风险模型下给出极大似然估计及渐近性,提出两种估计方法并比较二者表现,得到最优权估计更加有效.同时本文还考虑了半参数的风险模型,并基于鞅的估计方程得到其估计及渐近性,均得到不错的结果.     

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

Estimation of error variance in ANOVA model and order restricted scale parameters

We consider the estimation of error variance in the analysis of experiments using two level orthogonal arrays. We address the estimator which is the minimum of all the estimators which we obtain by pooling some sums of squares for factorial effects. Under squared error loss, we discuss whether or not this estimator uniformly improves upon the best positive multiple of error sum of squares. We show that when we have two factorial effects, we obtain uniform improvement. However, we show that when we have more than two factorial effects, we cannot necessarily obtain uniform improvement. Further, the above results are applied to the problem of estimating the smallest scale parameter of chi-square distributions.     

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.     

Exploring an unknown graph

We wish to explore all edges of an unknown directed, strongly connected graph. At each point, we have a map of all nodes and edges we have visited, we can recognize these nodes and edges if we see them again, and we know how many unexplored edges emanate from each node we have visited, but we cannot tell where each leads until we traverse it. We wish to minimize the ratio of the total number of edges traversed divided by the optimum number of traversals, had we known the graph. For Eulerian graphs, this ratio cannot be better than two, and two is achievable by a simple algorithm. In contrast, the ratio is unbounded when the deficiency of the graph (the number of edges that have to be added to make it Eulerian) is unbounded. Our main result is an algorithm that achieves a bounded ratio when the deficiency is bounded. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 265–297, 1999     

Asymptotic behaviour of a closed‐loop thermosyphon with linear friction and viscoelastic binary fluid

Viscoelastic fluids represent a major challenge both from an engineering and from a mathematical point of view. Recently, we have shown that viscoelasticity induces chaos in closed‐loop thermosyphons even when we consider binary fluids, this is, when we consider a solute in the fluid, as water and antifreezes, for example. In this work, we consider a linear friction law, and we show that in this case with the addition of a solute to the fluid we can prove, under some conditions, chaotic asymptotic behavior for suitable geometry of the circuit and heat flux or ambient temperature functions.     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

The value of foresight

Suppose you have one unit of stock, currently worth 1, which you must sell before time $T$. The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see $a$ units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresight.     

基于跳扩散过程的可转换债券的定价

本文标的股票的方程采用跳扩散方程,首先规定一个跳跃的涨跌区间,这样就可以很快的找出跳跃点,我们根据跳跃点将股价聚类,然后把各个类看成是总体中抽取出来的一个样本,我们就可以估计出跳扩散方程中的所有参数.由于我们的标的股票的方程是含跳过程,因此无法找出完全保值的自融资策略,但我们可以根据风险最小化的原理给出可转换债券的价格,最后运用Monte Carlo模拟计算出了南京水运转债在0时刻的价格。     

Forecasting the integration of immigrants

In this research, we develop and introduce a theoretical and mathematical forecasting framework of immigrant integration using immigrant density as a single driver. First, we introduce the integration concepts we aim at forecasting. Thereafter, we introduce a theoretical and mathematical model of the relationship between integration and immigrant density. Based on this model, we develop a methodological forecasting framework. We test the framework using immigrant integration data from Spain. We produce the forecasts, and conduct the proper evaluation of them. Finally, we conclude with a brief discussion of the wider implications of our results.