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.     

The structure factor for liquid lead

Abstract

The structure factor for liquid lead at the temperatures 613 K, 643 K, 863 K and 1163 K are measured by neutron diffraction. Extensive comparisons to earlier measurements are made as well as comparisons to different hard-core models.     

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.     

UV-Derivative Spectrophotometric Determination of Ritonavir Capsules and Comparison with LC Method

Abstract

An ultraviolet-derivative spectrophotometric method (UV-D) has been proposed as an alternative to a previously described liquid chromatographic (LC) method for the quantitative determination of ritonavir in soft gelatin capsules. The spectrophotometric method is based on recording the second-derivative spectra for ritonavir at 222.3 nm of its solutions in methanol. The linear dynamic range was 10.0–30.0 µg · mL^?1 with a correlation coefficient of 0.9995. Mean recoveries were between 99.2% and 100.2% for the tested capsules samples. Mean intra- and interassay relative standard deviations (RSDs) were less than 2.0%. The statistic analysis showed that LC and UV-D methods were equivalent to assay ritonavir capsules.     

Oxygen consumption and chemisorption in low-temperature oxidation of sub-bituminous pulverized coal

Studying the effect of oxygen in coal oxidation is very important for understanding and controlling coal spontaneous combustion. However, the oxygen effect is not very easy to determine clearly due to the large effect of heat source on coal oxidation in temperature rising experiments. Here, focused on sub-bituminous coal, the oxygen effect was separated from coal oxidation by continuously measuring FTIR spectra of coal with respect to varying temperatures and under oxygen and nitrogen. The active groups’ real-time changes of coal oxidation, thermal treatment and oxygen effect were measured. The carboxylic ester and carboxyl units are the main functional groups that increase with temperatures increasing under oxygen and nitrogen, while the other functional groups decrease in quantity. The oxygen effect promoted the consumption of aliphatic hydrocarbons and hydroxyl groups and also promoted the formation of oxygen-containing groups (except hydroxyl). Four characteristic temperature stages involved in the oxygen effect and their key functional groups were identified. Simultaneously, the relationship of oxygen consumption and chemisorption in oxygen effect was analyzed. The starting temperature of oxygen chemisorption is between 50 and 60°C. The maximum contribution of oxygen effect was observed in methyl and methylene groups. These results are important for chemical control of coal spontaneous combustion. The oxidation of aliphatic hydrocarbon should be controlled before oxygen chemisorption. The value of oxygen consumption between 70 and 80°C can be measured accurately due to the constant chemisorption rate, which help to identify the tendency for spontaneous combustion. These results will help in better understanding of the reaction mechanism of coal oxidation, especially the oxygen effect.     

Selectivity of seismic electric signal (SES) of the 2000 Izu earthquake swarm: a 3D FEM numerical simulation model

Although seismic electric signal (SES) has been used for short-term prediction of earthquakes, selectivity of SES still remains as one of the mysterious features. As a case study, we made a numerical simulation based on a 3D finite element method (FEM) on the selectivity of SES observed in the case of the 2000 Izu earthquake swarm. Our numerical results indicated that the existence of conductive channel under Niijima island could explain the reported SES selectivity.     

Rhorix: An interface between quantum chemical topology and the 3D graphics program blender

Chemical research is assisted by the creation of visual representations that map concepts (such as atoms and bonds) to 3D objects. These concepts are rooted in chemical theory that predates routine solution of the Schrödinger equation for systems of interesting size. The method of Quantum Chemical Topology (QCT) provides an alternative, parameter‐free means to understand chemical phenomena directly from quantum mechanical principles. Representation of the topological elements of QCT has lagged behind the best tools available. Here, we describe a general abstraction (and corresponding file format) that permits the definition of mappings between topological objects and their 3D representations. Possible mappings are discussed and a canonical example is suggested, which has been implemented as a Python “Add‐On” named Rhorix for the state‐of‐the‐art 3D modeling program Blender. This allows chemists to use modern drawing tools and artists to access QCT data in a familiar context. A number of examples are discussed. © 2017 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

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.     

A strategy for multivariate calibration based on modified single-index signal regression: Capturing explicit non-linearity and improving prediction accuracy

In this paper, a modified single-index signal regression (mSISR) method is proposed to construct a nonlinear and practical model with high-accuracy. The mSISR method defines the optimal penalty tuning parameter in P-spline signal regression (PSR) as initial tuning parameter and chooses the number of cycles based on minimizing root mean squared error of cross-validation (RMSECV). mSISR is superior to single-index signal regression (SISR) in terms of accuracy, computation time and convergency. And it can provide the character of the non-linearity between spectra and responses in a more precise manner than SISR. Two spectra data sets from basic research experiments, including plant chlorophyll nondestructive measurement and human blood glucose noninvasive measurement, are employed to illustrate the advantages of mSISR. The results indicate that the mSISR method (i) obtains the smooth and helpful regression coefficient vector, (ii) explicitly exhibits the type and amount of the non-linearity, (iii) can take advantage of nonlinear features of the signals to improve prediction performance and (iv) has distinct adaptability for the complex spectra model by comparing with other calibration methods. It is validated that mSISR is a promising nonlinear modeling strategy for multivariate calibration.     

A novel clipping technique for filtering FM signals embedded in intensive noise

A novel adaptive clipping technique for filtering a constant amplitude frequency modulated (FM) signal embedded in non-Gaussian noise is proposed. It is based on the analysis and processing of the estimate of probability density function of a FM signal realization. As a result, modifications of two robust estimators of FM signal amplitude are proposed. It is shown that these estimators can be used for Gaussian and non-Gaussian heavy-tail environments. The proposed clipping technique can exploit one or another obtained robust estimate of the signal amplitude for adaptive setting a threshold. Analysis of signal estimate accuracy for different noise environments is carried out. Comparative analysis of the obtained methods and known approaches based on scanning window nonlinear filtering and optimal robust L-DFT form is performed. It is demonstrated that the usage of clipping-based technique leads to the considerable improvement of the FM signal filtering efficiency in comparison to the aforementioned known approaches for different noise environments and a wide range of input SNR values.