Existence and uniqueness of periodic solutions for a class of nonlinear n ‐th order differential equations with delays

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T ‐periodic solutions for a class of nonlinear n ‐th order differential equations with delays of the form x⁽ⁿ⁾(t) + f (x^{(n‐ 1)}(t)) + g (t, x (t ‐ τ (t))) = p (t).     

Inversion of degree <Emphasis Type="Italic">n</Emphasis> + 2

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C _n ¹ and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in the Euclidean space the pedal surfaces of the congruences C _n ¹ can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.      

Chromatography: The separation technique of the 20th century

Summary M. S. Tswett contemplated the possibility of “chromatography” in 1899–1901 while carrying out his first research work on the physico-chemical structure of plant chlorophylls, and he reported “on a new category of adsorption analysis” in 1903. The evolution of chromatography followed the advances of this century: each decade brought new innovations based logically from the previous one. By the end of the 20th Century chromatography has became the most widely used separation technique in chemistry and biochemistry: thus, it is no exaggeration to call it the separation technique of the 20th Century. This paper investigates the evolution of the various branches of chromatography. An essential condition for all fruitful research is to have at one’s disposal a satisfactory technique. “Tout progrès scientific est un progrès de méthode” as somebody once remarked. Unfortunately the methodology is frequently the weakest aspect of scientific investigations.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

多模辐射场的广义非线性不等阶高阶压缩的一般理论

本文建立了多模辐射场的广义非线性不等阶高阶压缩的一般理论。首次提出了多模辐射场中各模压缩阶数不相等的N_j次方Y压缩、N_j次方H压缩以及N_j次方X压缩的一般性定义,并对N_j-Y最小测不准态、N_j-Y压缩最小测不准态、N_j-H最小测不准态、N_j-H压缩最小测不准态以及N_j-X最小测不准态和N_j-X压缩最小测不准态等进行了详细讨论。指出,Zhang等人在国际上提出的有关单模辐射场的振幅N次方压缩的定义、Mark Hilery在国际上提出的有关双模辐射场的“双模和压缩”与“双模差压缩”的定义以及由笔者新近提出的有关多模辐射场的N次方Y压缩、N次方H压缩和N次方X压缩等的定义,仅仅只是本文所提出的N_j次方Y压缩、N_j次方H压缩以及N_j次方X压缩等的一般性定义在各种不同条件下的特例.     

南开大学分析化学学科的建设与发展

2021年正值南开大学化学学科创建100周年。作为南开化学的一个重要分支,南开大学分析化学学科始建于1952年。回顾了南开分析学科的历史脉络和发展成就;着重介绍了南开分析学科的发展现状;面向国家重大需求,提出了学科的未来发展方向。谨以此文,敬献给为南开分析学科发展做出贡献的南开人及支持、帮助本学科发展的朋友们！向南开大学化学学科创建100周年致敬！     

高准确度光电成像测量系统图像畸变校正算法

在以CCD为核心器件的光电成像测量系统对物体进行线度测量中,成像光学系统的固有特性使得数字图像存在畸变,根据像差理论建立了三次多项式的畸变校正模型,分析比较了不同的灰度重建方法的优缺点,进行了双线性插值的灰度重建.对一种校正图像进行了噪音点滤除,进行了仿真实验.校正后的图像可用于高准确度的成像测量.