高温作业专用服装设计

建立了热防护服系统的偏微分方程模型,给出了适当的初始条件、边界条件及交界面条件,采用隐式差分格式进行计算.在此基础上,建立了热防护服最优参数设计的优化模型,采用二分法求解.最后对2018年"高教社杯"全国大学生数学建模竞赛A题的参赛论文予以评述.     

两个同心旋转圆柱之间的两种流体的交界面几何形状问题

研究两个同心旋转圆柱之间的两种流体的交界面几何形状问题.利用张量分析工具,给出了忽略耗散能量影响下交界面几何形状是一种能量泛函的临界点,其对应的Euler-Lagrange方程是1个非线性椭圆边值问题.对于粘性引起的耗散能量不能忽略的情况下,同样给出了1个带有耗散能量的能量泛函,其临界点是交界面几何形状,相应的Euler-Lagrange方程也是1个二阶的非线性椭圆边值问题.这样,交界面几何形状问题转化为求解非线性椭圆边值问题.     

交界面椭圆、抛物和Maxwell方程的有限元分析和进展 献给林群教授80华诞

本文综述交界面问题的有限元离散数值方法近十几年来的最新进展,重点讨论椭圆型、抛物型和Maxwell型交界面问题的有限元方法,总结已有的相关研究结果,如正则性、变分形式、有限元离散策略、数值积分策略、交界面逼近及其对离散解的影响和离散解的收敛性分析等.伴随历史发展,本文还梳理了交界面问题的发展脉络.最后列举一个交界面模型的实际应用,即不可测热交换界面上热通量重构问题,着重分析自适应有限元方法的后验误差估计.     

血液中酒精浓度的数学模型

把人体对酒精的吸收、排放简化为一般的房室模型,提出了吸收因子、消除因子的概念.针对短时间饮酒、长时间饮酒以及间断饮酒等情况,分别建立了关于人体体液中酒精浓度的微分方程模型,并且给出了显式解.对于特殊的周期性间断饮酒的模型,给出了更便于计算的叠加公式,并通过分析酒精浓度函数的极限过程,证明了其有界性.对短时间饮酒和长时间饮酒的情况分别计算了酒精浓度的最大值、取得最大值的时间和禁止驾车的时间范围,而且进行了比较,所得结论与实际吻合.     

具边界相交交界面的拟线性椭圆方程组的挠射问题

研究有界区域上的n维拟线性椭圆方程组的挠射问题,其中方程的系数在交界面上允许间断,而且交界面允许与区域外边界相交.通过构造一个交界面与外边界不相交的近似挠射问题,利用估计和逼近方法,并讨论弱解的正则性,得到了问题解的存在性,并将这些结果应用到两种群Lotka-Volterra互惠模型.     

具广义非线性源的波动方程的高能爆破

研究一类具广义非线性源的非线性波动方程的初边值问题在高初始能级状态下解的有限时间爆破.利用经典的凹函数方法找到了导致该问题具任意正初始能级的解有限时间爆破的初值.     

非均质地层中自然电位测井的数学模型和数值方法

针对非均质地层中电阻率非分块常数,提出一个新的自然电位测井模型.自然电位满足具有交界面跳跃的椭圆边值问题.证明了该椭圆边值问题弱解的存在唯一性;提出一种捕捉交界面电位跳跃的有限差分方法求解此类问题,数值例子验证了该方法的可靠性和有效性.     

自适应的Dantzig选择器的渐近性质研究

本文首先研究当变量个数p较大、甚至关于样本n指数速度增长时,高维稀疏线性回归模型下适应的Dantzig选择器估计量的渐近性问题.作为适应的Dantzig选择器的权重,当回归系数的初始估计取为某常数的相合估计时,在一些常规条件下,本文证明了适应的Dantzig选择器具有Oracle性质.对于p≤n和pn两种情形,本文也分别给出了易于实现的初始估计量.最后通过模拟验证了前面的理论结果.     

双层Kidder自相似解及其Rayleigh-Taylor不稳定性研究

将单层Kidder自相似解推广到双层,使得两层壳体的交界面两侧存在密度跳跃,使得轻流体向重流体加速产生Rayleigh-Taylor不稳定性;通过采用Lagrange坐标下的Godunov方法进行一维直接数值模拟,将模拟解与双层Kidder自相似基本解进行比较,验证了双层Kidder自相似解的可靠性;最后,通过编制球形内爆的三维扰动的线性稳定性分析程序,对双层Kidder自相似解的Rayleigh-Taylor不稳定性进行了分析计算.计算结果表明:初始扰动越集中于交界面,会造成后期扰动增长得越快,越不稳定;扰动波数越大,扰动增长得越快,越不稳定;从扰动在空间上的发展来看,可压缩性研究表明内外壳体的可压缩性对扰动增长起着相反的作用,外层壳体的可压缩性对Rayleigh-Taylor不稳定起失稳作用,而内层壳体的可压缩性对Rayleigh-Taylor不稳定起致稳作用.     

饮酒驾车的数学模型

利用药物代谢动力学方法,结合拟合与迭加,借助Matlab 6.5程序,较准确方便地求出了短时间饮酒、长时间饮酒及天天饮酒情形下血液中酒精浓度变化关系式以及达到峰浓度和新安全标准浓度的时间关系式,由此,对2004“高教社杯”全国数学建模竞赛C题中的各问题给出了完整的解答.     

Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

In this paper, we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation $$u_t−{\rm div}(|∇u^m|^{p−2}∇u^m)=0$$ in $\mathbb{R}^N$ with $p>1,$ $m>0$ and $m(p−1)−1>0.$ By using the finite propagation property and the $L^1-L^∞$ smoothing effect, we find that the complicated asymptotic behavior of the rescaled solutions $t^{\mu/2}u(t^{β_·},t)$ for $0<\mu<2N/(N[m(p−1)−1]+p)$ and $β>(2−\mu[m(p−1)−1])/(2p)$ can take place.     

Sufficient Conditions for Starlikeness and Strong-starlikeness of a Given Order

We find sharp upper bounds for $ \vert \,f^{\prime \prime }(z)/f^{\prime }(z) \vert $ such that the function f be starlike or strongly-starlike of a given order, where f is holomorphic in the unit disc U , with $ f(0)=f^{\prime }(0)-1=0 $ . These bounds are obtained by using the differential subordination technique. As a useful ingredient we obtain the radius of convexity for the function $ F(z)= z /(\exp (z)-1) $     

Characterizations of Bochner–Krall Orthogonal Polynomials of Jacobi Type

Let $\{P_n(x) \}_{n=0}^\infty$ be an orthogonal polynomial system relative to a compactly supported measure. We find characterizations for $\{P_n(x) \}_{n=0}^\infty$ to be a Bochner--Krall orthogonal polynomial system, that is, $\{P_n(x) \}_{n=0}^\infty$ are polynomial eigenfunctions of a linear differential operator of finite order. In particular, we show that $\{P_n(x) \}_{n=0}^\infty$ must be generalized Jacobi polynomials which are orthogonal relative to a Jacobi weight plus two point masses.     

A Relaxed Approximate Proximal Point Algorithm

For a maximal monotone operator T, a well-known overrelaxed point algorithm is often used to find the zeros of T. In this paper, we enhance the algorithm to find a point in , where is a given closed convex set. In the inexact case of our modified relaxed proximal point algorithm, we give a new criterion. The convergence analysis is quite easy to follow.     

Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization

We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.     

一种对称损失函数下刻度参数的最优同变估计

本文研究刻度参数分布族(1/σ)f(x/σ)中刻度参数在损失函数L(σ,δ)=(σ-δ)~2/σδ下的最小风险同变估计及其最小最大性。     

Self-dual codes over and half-integral weight modular forms

In this paper, we find a connection between the weight enumerator of self-dual codes and half-integral weight modular forms. We generalize in that way results of Broué-Enguehard, Hirzebruch, Ozeki, Rains-Sloane, Runge.

     

给定零点的最近复系数多项式计算问题

Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f(z) and a zero α, in this paper we explore the problem of computing a complex polynomial f(z) such that f(α) = 0 and the distance ∥f-f ∥ is minimal. Considering most of the existing works focus on either certain polynomial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of f(z), we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f(z) explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.     

Davey-Stewartson系统粗糙爆破解的极限行为

研究了Davey-Stewartson系统(简记为D-S系统)粗糙爆破解的动力学性质.所谓粗糙爆破解即为正则性为H~s(s1)的爆破解,此时D-S系统粗糙解不再满足能量守恒率.利用I-方法与Profile分解理论,得到了D-S系统粗糙爆破解在H~s(R~2)(其中ss_0,且s_0≤(1+11~(1/2))/5≈0.8633)中的极限行为,包括L~2强极限的不存在性与L~2集中性质以及极限图景.     

Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series

Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of ; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter is the eigenvalue of ) for a cubic phase.

As applications we find sets of asymptotic relations for divisor functions.