数学建模对高中生构建数学底层思维的作用及其教学实施建议

数学底层思维即用数学的眼光观察世界、用数学的思维分析世界以及用数学的语言表达世界,是人们面对自然和社会中纷繁多样的现象和问题时,所展现的自发的、不依赖监督的、融汇数学学科核心素养的思维方式.作为国家高中新课程标准中数学六大核心素养之一的数学建模,是培养学生数学底层思维的良好载体,对人才培养和社会发展均起到良好的促进作用.本文主要阐述了数学建模对高中生构建数学底层思维的作用,并结合教学实例给出教学实施建议.     

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near , the uniform asymptotic expansion involves Airy function as , and Bessel function of order α as in the neighborhood of , the uniform asymptotic expansion is associated with Bessel function of order β as . The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.     

Mathematical Modeling of Nylon 6/6,6 Copolymerization in Batch Reactor: Investigating Recipes without Water and with Cyclic Dimer

A model is used to simulate batch copolymerization of caprolactam with hexamethylene diamine (HMD) and adipic acid (ADA) to produce nylon 6/6,6. Four different recipes are considered: a recipe containing caprolactam and an aqueous solution of HMD and ADA, a recipe containing caprolactam and dry HMD/ADA salt, and two recipes with a portion of the caprolactam replaced by nylon 6 cyclic dimer (CD). Consuming CD would be advantageous because CD is an undesirable side product from nylon 6 production. Simulation results lead to three important findings: (i) operation using dry salt rather than aqueous salt solution leads to higher degree of polymerization, (ii) substantial quantities of CD can be consumed to produce nylon 6/6,6 copolymer, and (iii) including water in the recipe is beneficial for achieving improved consumption of CD. The results of this study will be helpful in designing experiments aimed at improving industrial nylon 6/6,6 copolymerization processes.     

A differential-equation-based model of the glass ceiling in career progression

ABSTRACT

We introduce a model based on Ordinary Differential Equations to describe how two mutually exclusive groups progress through a career hierarchy, whether in a single organization, or in an entire economic sector. The intended application is to gender imbalance at the top of the academic hierarchy in European Universities; however, the model is entirely generic and may be applied in other contexts also. Previous research on gender imbalance in European universities has focused on large-scale statistical studies. Our model represents a point of departure, as it is deterministic (i.e., based on Ordinary Differential Equations). The model requires a precise definition of the progression rates for the different groups through the hierarchy; these are key parameters governing the dynamics of career progression. The progression rate for each group can be decomposed into a product: the proportion of group members at a low level in the hierarchy who compete for promotion to the next level a given year, multiplied by the in-competition success rate for the group in question. Either of these two parameters can differ across the groups under consideration; this introduces a group asymmetry into the organization’s composition. We introduce a glass-ceiling index to summarize this asymmetry succinctly. Using case studies from the literature, we demonstrate how the mathematical framework can pinpoint the proximate cause of the glass ceiling in European academia.     

DIOPTAS: a program for reduction of two-dimensional X-ray diffraction data and data exploration

The amount of data collected during synchrotron X-ray diffraction (XRD) experiments is constantly increasing. Most of the time, the data are collected with image detectors, which necessitates the use of image reduction/integration routines to extract structural information from measured XRD patterns. This step turns out to be a bottleneck in the data processing procedure due to a lack of suitable software packages. In particular, fast-running synchrotron experiments require online data reduction and analysis in real time so that experimental parameters can be adjusted interactively. Dioptas is a Python-based program for on-the-fly data processing and exploration of two-dimensional X-ray diffraction area detector data, specifically designed for the large amount of data collected at XRD beamlines at synchrotrons. Its fast data reduction algorithm and graphical data exploration capabilities make it ideal for online data processing during XRD experiments and batch post-processing of large numbers of images.     

An Improved Model for Polyether Production from 1,3‐Propanediol

A dynamic mathematical model is developed for production of Cerenol polyether from 1,3‐propanediol in a batch reactor system. The model accounts for polycondensation reactions and side reactions in the liquid phase and for mass transfer of volatile species to the vapor. Parameters are estimated using measured liquid‐phase concentrations of monomer, oligomers, water, and end groups as well as the mass and composition of condensate collected from the overhead condenser system. The proposed model uses novel probability factors to keep the model equations relatively simple while accounting for the complex influence of superacid catalyst on reaction rates. The model is a significant advance over previous Cerenol models because it better accounts for mass‐transfer rates and for the dynamic behavior of the condenser. In addition, the proposed model accounts for the inhibitory influence of water on polycondensation kinetics due to hydration of hydroxyl ends. The model equations and parameter estimates provide a substantial improvement in fit to the data, especially for long reaction times and high catalyst levels, resulting in a 97% reduction in the value of the weighted least squared objective function compared to equations and parameters from a previous model.     

EAST极紫外空间分辨光谱分析软件的开发

基于Matlab软件的图形用户界面开发环境(GUIDE)设计开发了一套交互式界面，可以自动读取、解析原始数据，开展波长标定，将谱强度与谱仪系统空间测量位置以及EAST实验放电时间等信息相关联，从而快速、便捷地计算杂质离子发出的线光谱强度的空间分布并绘图。该空间分辨光谱分析软件的开发和应用大大提高了谱仪数据处理和分析的便捷性和效率，保证了放电期间数据处理的准实时性。     

Some physico-chemical characteristics of amino-acid derivatives containing rest of 1,2,4-triazole-3,4-di-substituted with potential antitumoral activity

Abstract

Dipeptide derivatives from p-aminobenzoyl-phenylglycine and p-aminobenzoyl-phenylalanine were grafted on 1,2,4-triazole heterocycle in order to get new substances with potential biological activity. The compounds synthetized and characterized in this paper have a structure that fits in the antimetabolites category, products used to treat malignant tumors. The presence of several bioactive groups in the same molecule permits us to study the chemical structure - biological activity correlation based on computational analyses using SPARTAN’14 software. Some correlations between the molecular parameters established with Spartan’14 software are evidenced in this paper.     

Various solution methods,accompanied by dynamic investigation,for the same problem as a means for enriching the mathematical toolbox

A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various branches of mathematics and to highlight the importance of providing the students with an extensive ‘mathematical toolbox’. Investigation of the property that appears in the task was carried out using a computerized tool.