Computational Construction as a Means to Coordinate Representations of Infinity

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

On the well-posedness of a two-phase minimization problem

We prove a series of results concerning the emptiness and non-emptiness of a certain set of Sobolev functions related to the well-posedness of a two-phase minimization problem, involving both the p(x)-norm and the infinity norm. The results, although interesting in their own right, hold the promise of a wider applicability since they can be relevant in the context of other problems where minimization of the p-energy in a part of the domain is coupled with the more local minimization of the L^∞-norm on another region.     

On the dimension of trees

We consider isometric embedding of trees into the infinite graph Z_m whose vertices are the m-dimensional lattice points where two vertices a=(a₁,a₂,…,a_m) and b=(b₁,b₂,…,b_m) are adjacent if and only if |a_i-b_i|?1 for 1?i?m. Linial, London, and Rabinovich have shown that this can be done with , where t is the number of leaves. In this note, we sketch a proof that .     

Escape dynamics in collinear atomic-like three mass point systems

The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise interaction. Specifically, we focus on the asymptotic behaviour for systems with non-negative total energy.On the zero energy level set there are two distinct asymptotic states, called 1+1+1escape configurations, where all the three separations infinitely increase as t→∞. We show that 1+1+1 escapes are improbable by proving that the set of initial conditions leading to such asymptotic configurations has zero Lebesgue measure. When the outer mass points are of the same kind we deduce the existence of a heteroclinic orbit connecting the 1+1+1 escape configurations. We further prove that this orbit is stable under parameter perturbation.In the positive energies’ case, we show that the set of initial conditions leading to 1+1+1 escape configurations has positive Lebesgue measure.     

一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量,然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

On the extra-column band-broadening contributions of modern, very high pressure liquid chromatographs using 2.1 mm I.D. columns packed with sub-2 μm particles

The efficiencies of two narrow bore columns (100 mm and 50 mm × 2.1 mm) packed with 1.7 μm totally porous BEH-C₁₈ particles were measured on two very high pressure liquid chromatographs (Acquity from Waters and 1290 Infinity HPLC System from Agilent) operating at maximum pressures of 1034 and 1200 bar, respectively. The probe compounds were a mixture of uracil, acetophenone, toluene, and naphthalene eluted in a 50/50 (v/v) solution of acetonitrile and water at 303 K with a flow rate of 0.40 mL/min. The apparent efficiencies of columns, which lumps the consequences of band broadening due to the column and the system contributions, may depend much on the extra-column volumes of the instruments used. Actually, it is known for a long time that the apparent column performance is strongly affected by the instrument characteristics, including the diameter of the connecting tubes, the injection technique (with or without needle seat capillary), and the detection cell volume. When the 1290 Infinity HPLC System is equipped with a needle seat, an inlet and an outlet connecting capillary tube with inner diameters around 115 μm, its extra-column variance for a 0.1 μL injection volume is 9.2 μL² while that of the Acquity instrument is 6.9 μL². Minor modifications suggested by their respective manufacturers allowed significant reductions of these variances, to 6.2 and 3.9 μL², respectively. Yet, in their optimized configurations and for weakly retained compounds (k ? 1), these modern, sophisticated instruments cannot provide more than 75% (1290 Infinity) and 85% (Acquity) of the maximum efficiency of a 2.1 mm × 50 mm BEH column. For more strongly retained compounds (k > 4), in contrast, they are both able to provide more than 95% of the maximum expected efficiency.     

全能近似分析简介

根据无限的“无有终了”的事实,应当把无尽小数看作无穷数列简写,采用这种观点就可以得到实数的运算法则;如果用康托（Cantor,G）的“无限是现实的、完成了的、存在着的整体”的“实无限”观点就得不到这个法则．点是针对误差界的足够小,其中没有大小的点叫做理想点,有大小的点叫做近似点．理想点具有无法点出的性质．近似点的集合能够组成线段,但理想点的集合不能组成线段．绝对准确地讨论没有大小的理想瞬时上的速度没有实际意义,理想的瞬时速度依赖于近似瞬时的速度．对于点、线、面、实数、函数、导数、积分、积分变换、实数集等数学名词都需要提出近似、理想、全能近似三类技术术语,应用对立统一法则去阐述数学理论．     

Inhomogeneous infinity Laplace equation

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously.