关于全国大学生数学竞赛决赛试题的解析与思考

本文对二道全国大学生数学竞赛决赛试题(数学类)给予了诠释与解析.     

Computer Assisted Spectrophotometric Analysis of Three Component Mixtures (Caffeine Mixtures and Codeine Mixtures)

Abstract

Two mixtures (caffeine mixture (I) and codeine mixture (II)) of 3-components, containing acetaminophen and salicylamide as majors, have been assayed using different computer-assisted spectrophotometric methods. These methods are the Unique Absorbance Method (UAM), the Unique Orthogonal Function Method (UOFM), and Unique Fourier Function Method (UFFM), and also their correspondences under the least squares approach. The parameters have been optimized for the accurate determination of the minor component caffeine in mixture (I). In mixture (II), however. codeine has been extracted - in two steps-prior to the correction of interferences from the major components acetaminophen and salicylamide.     

A new class of multi-wavelet bases: V-system

The V-system is a complete orthogonal system of functions defined on the interval [0, 1], generated by finite Legendre polynomials and the dilation and translation of a function generator, which consists of a finite number of continuous and discontinuous functions. The V-system has interesting properties, such as orthogonality, symmetry, completeness and short compact support. It is shown in this paper that the V-system is essentially a special multi-wavelet basis. As a result, some basic properties of the V-system are established through the well-developed theory of multi-wavelets. From this point of view, more other V-systems are constructed.     

主链型阳离子表面活性剂聚季铵盐的制备

以N,N-二甲基-1,3-丙二胺、尿素及1,4-二氯丁烷为原料,通过共缩聚反应两步法制备了一种新的主链型的水溶性聚季铵盐.采用正交试验获得了制备聚季铵盐的最佳条件：N,N-二甲基-1,3-丙二胺与尿素的物质的量比为2∶1,反应温度为135℃,反应时间为11h,得到中间体双[3-（N,N-二甲基丙胺基）]丙脲,产率高达99％;中间体与1,4-二氯丁烷的物质的量比为1∶1,于90℃下反应6h.采用红外光谱和核磁氢谱对产物结构进行表征,测定了其热稳定性和特性粘数.     

阴离子表面活性剂聚二硫二丙烷磺酸钠的合成

利用硫化钠与硫磺反应制备二硫化钠,然后将二硫化钠与1,3-丙磺酸内酯反应,合成了一种可作为电镀添加剂的阴离子表面活性剂——聚二硫二丙烷磺酸钠(SPS)。采用核磁共振氢谱对合成产物进行结构表征,确认了产物结构及产率。通过正交试验研究了产物产率与反应物配比、反应温度、溶剂加入量等因素之间的关系,找出了最优合成条件:第一步合成二硫化钠的反应中,硫化钠/硫磺物质的量比为1∶1.3,温度55℃,加水量18 mL;第二步合成SPS的反应中,1,3-丙磺酸内酯/硫化钠物质的量比为1.7∶1,温度40℃,溶剂量75 mL,产物的最高产率可达到95.8%。     

“蓝瓶子”实验再探究*

利用SPSS 25.0统计软件对“蓝瓶子”实验进行正交实验设计,通过统计学分析得出最佳实验条件。利用氧化还原传感器(ORP)对“蓝瓶子”振荡体系的电极电势变化曲线进行分析,并针对蓝色的产生和消失过程设计3组对比实验深入探究,阐明其反应原理,意在引导学生深刻理解趣味实验现象背后的微观反应本质。     

确定性矩阵可分离压缩成像

针对可分离压缩传感使用的可分离随机正交矩阵在处理大尺度图像等高维信号感知时难度太大或成本过高的问题,引入确定性测量矩阵,提出确定性矩阵可分离压缩传感,可将如托普利兹矩阵及循环矩阵等具有确定性结构的矩阵作为可分离压缩传感的左、右可分离矩阵.该方案可以降低独立元素的数目,从而显著降低前端物理实现的难度与成本.数值模拟实验分别评估了该方法在不同采样率及不同图像尺寸下的压缩重建性能,结果表明该方法在独立元素非常少的情形下得到与原随机正交矩阵相近的重建质量,证明了其可行性.     

A Three‐Factor Product Construction for Mutually Orthogonal Latin Squares

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, “triple products” of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.     

Research on improving the accuracy of near infrared non-invasive hemoglobin detection

To optimize the accuracy of near-infrared non-invasive hemoglobin (Hb) clinical detection, high-performance instrument and prepossessing algorithm have been investigated. A near-infrared spectrophotometric system was constructed adopting InGaAs detector array with 16 pixels and plane grating spectrometer to obtain high signal noise ratio (SNR) spectral data. In our experiment, we applied the device independently to collect spectra data from 91 volunteers’ fingertips non-invasively. Two prediction tests were conducted to verify the effects of preprocessing algorithms improving the accuracy of near-infrared Hb detection and exclude the occasionality of satisfactory results in a single trial. Our non-invasive Hb detection methods were based on partial least squares (PLS). In each test, PLS, MSC coupled with PLS, DOSC coupled with PLS, three methods for non-invasive Hb detection, were analyzed respectively. The results of two trials showed that only DOSC & PLS performed excellently in both predictive ability and stability, obviously better than other two methods. Relative RMSEP was 6.16% in predicting test 1, 6.08% in predicting test 2, almost reaching the requirements of clinical application. It indicates that our independent-developed high-performance instrument and the method DOSC coupled with PLS are promising in non-invasive Hb detection clinical application.