Diameters of complete sets of conjugate algebraic integers of small degree

We give bounds for the coefficients of a polynomial as functions of the diameter of its roots, hence we obtain polynomials with minimal diameters and small degree

     

求代数曲线斜渐近线的一种新方法

根据对渐近线的定义的理解,给出了求代数曲线斜渐近线的一种新方法．     

二阶代数微分方程超越亚纯解

Using Nevanlinna theory of the value distribution of meromorphic functions,we discuss some properties of the transcendental meromorphic solutions of second-order algebraic differential equations,and generalize some results of some authors.     

Jacobi spectral solution for integral algebraic equations of index-2

This paper is concerned with obtaining the approximate solution of a class of semi-explicit Integral Algebraic Equations (IAEs) of index-2. A Jacobi collocation method including the matrix-vector multiplication representation is proposed for the IAEs of index-2. A rigorous analysis of error bound in weighted L² norm is also provided which theoretically justifies the spectral rate of convergence while the kernels and the source functions are sufficiently smooth. Results of several numerical experiments are presented which support the theoretical results.     

Numerical stroboscopic averaging for ODEs and DAEs

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt=f(y,t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt=F(Y) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step-sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LIPS algorithm of Kirchgraber and multirevolution methods. An error analysis is provided that indicates that the efficiency of all these techniques increases even further when combined with splitting integrators.     

一种确定求积公式误差最优估计的简单方法

利用求积公式代数精度的概念,给出一种确定Newton-Cotes和Hermite插值型求积公式截断误差最优估计的简单方法,并通过实例验证其有效性.     

线性规划单纯形代数7小步法与简易矩阵表格法

在《运筹学》这门课的教学过程中,单纯形法一直是教学的一个难点,学生也比较难理解、不容易学明白.通过多年的运筹学教学经验,针对目标为max的线性规划问题,提出"单纯形代数7小步法"和"简易矩阵表格法".对于"单纯形代数7小步法",只需要按照这7个步骤一步一步操作就能得到最优解和目标函数最优值;对于"简易矩阵表格法",根据题目的模型得到初始矩阵表格后,就是不断地在矩阵表格中寻找主元,然后将主元变成1,并将主元所在列的其他元素变成0,再根据矩阵的最后一行元素的正负进行最优性检验;最后得到最优矩阵表格,从最优矩阵表格里就能直接读出最优解和目标函数的最优值.将单纯形法提炼成比较容易理解和接受的这两种形式,为学生学习单纯形法提供重要的参考,同时也为运筹学老师的对这一部分内容的教学提供借鉴.     

关于高阶代数微分方程组亚纯解的计数函数（英文）

Using Nevanlinna theory and value distribution of meromorphic functions and the other techniques,we investigate the counting functions of meromorphic solutions of systems of higher-order algebraic differential equations and obtain some results.     

Generic Representation Theory of the Unipotent Upper Triangular Groups

It is generally believed (and for the most part it is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this article we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups U_n), and under a certain hypothesis relating the characteristic p to both n and the dimension d of a representation (specifically, p ≥ max(n, 2d)), Lie theory is completely sufficient to determine the representation theories of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their “generic” representation theory in characteristic p.     

Presenting higher stacks as simplicial schemes

We show that an n$n$-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n$n$-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n$n$-stacks, Deligne–Mumford n$n$-stacks and n$n$-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived n$n$-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich’s dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.