2002年中国数学奥林匹克试题解答

1　△ＡＢＣ的三边长分别为ａ ,ｂ ,ｃ ,ｂ <ｃ,ＡＤ是∠Ａ的平分线 ,点Ｄ在边ＢＣ上 ,1)求在线段ＡＢ ,ＡＣ内分别存在点Ｅ ,Ｆ(不是顶点 )满足ＢＥ =ＣＦ和∠ＢＤＥ =∠ＣＤＦ的充要条件(用角Ａ ,Ｂ ,Ｃ表示 ) ;图 1　题 1图2 )在点Ｅ和Ｆ存在的情况下 ,用ａ ,ｂ ,ｃ表示ＢＥ的长 .解　 1)设∠ＦＤＣ =∠ＥＤＢ =α ,则在△ＤＦＣ中 ,由正弦定理得ＣＦｓｉｎα =ＣＤｓｉｎ∠ＤＦＣ =ＣＤｓｉｎ(α +Ｃ) .即　　ＣＦ =ＣＤｓｉｎαｓｉｎ(Ｃ +α) (1)在△ＤＥＢ中 ,同理有　　　ＢＥ =ＤＢｓｉｎαｓｉｎ(Ｂ +α) (2 )由 (1) ,(2 )及ＢＥ …     

Exponentially Expanded Grid Network Approach （EEGNA） ： An Efficient Way for the Simulation of Stiff Electrochemical Problems

The exponentially expanded space grid was incorporated into the network approach to overcome the problem of low simulation efficiency during the simulations of electrochemical problems with stiff kinetics or wide dlsperslon of diffusion coefficients, resulting in an effective electrochemical simulation method: exponentially expanded grid network approach (EEG-NA). The stability and accuracy of the EEGNA for the simulation of various electrode processes .coupled with different types of homogeneous reactions were investigated.     

一种基于遗传算法的Job Shop作业排序方法

用遗传算法对ＪｏｂＳｈｏｐ零件排序问题进行优化搜索，并在算法中引Ａ了一些新的思想，以有利于降低种群的规模，提高计算速度，改善优化结果。     

煤油热沉与热裂解反应特性的试验研究

本文设计了一种煤油热裂解反应特性与热沉测量试验系统,利用该试验系统和原始标定法的热损测量方法,测量出了在亚临界和超临界压力下煤油热裂解反应的热沉大小,并进行了热裂解反应主要分解产物成分含量的测定与分析。结果表明：壁温随热流密度增加而增加;在相同燃油温度条件下,压力增大,热沉降低;裂解反应气体的浓度变化主要受到燃油温度和...     

火焰原子吸收光谱法测定四大怀药中10种元素

对四大怀药(即怀山药、怀地黄、怀菊花及怀牛膝)中的8项微量元素(铜、铁、锌、锰、镍、钴、铬和铅)、2种常量元素钾及钙,用火焰原子吸收光谱法做了测定。比较了水煎酸化和硝酸-高氯酸(4+1)混合酸直接消解两种溶样方法,结果表明:用后一种溶样方法可使上述10种元素从药材中更完全地提取。在所述的最优分析条件下,上述10种元素的质量浓度与其相应吸光度在一定范围内呈线性关系。对方法的回收率和精密度做了试验,测得回收率在97.0%～105.0%之间,相对标准偏差(n=5)均小于5%。     

聚酰亚胺涂覆的光纤宏弯温度传感器

为了提高光纤宏弯温度传感器的性能,提出了一种基于聚酰亚胺(PI)涂覆的新颖光纤宏弯温度传感器。利用基于纤芯-包层-无限涂覆层结构的光纤弯曲损耗-温度测量方法确定了传感器的光纤弯曲半径,将PI薄膜涂覆在1060-XP光纤包层外获得了新型的光纤宏弯温度传感器。该传感器的温度传感实验结果表明,PI涂覆不仅能提升光纤的机械性能和耐热性,还可实现温度灵敏度和温度测量分辨能力的显著提高。该新颖的光纤宏弯温度传感器可实现-20～100℃的宽温测量范围,温度灵敏度为0.072 dB/℃,分辨能力为0.14℃。与其他光纤宏弯温度传感器相比,所设计的传感器的温度传感性能显著提高。     

Molecular dynamics simulations of mechanical properties of epoxy-amine: Cross-linker type and degree of conversion effects

Molecular dynamics (MD) simulations are conducted to study the thermo-mechanical properties of a family of thermosetting epoxy-amines. The crosslinked epoxy resin EPON862 with a series of cross-linkers is built and simulated under the polymer consistent force field (PCFF). Three types of curing agents (rigidity1,3-phenylenediamine (1,3-P), 4,4-diaminodiphenylmethane (DDM), and phenol-formaldehyde-ethylenediamine (PFE)) with different numbers of active sites are selected in the simulations. We focus on the effects of the cross-linkers on thermo-mechanical properties such as density, glass transition temperature (T_g), elastic constants, and strength. Our simulations show a significant increase in the T_g, Young's modulus and yield stress with the increase in the degree of conversion. The simulation results reveal that the mechanical properties of thermosetting polymers are strongly dependent on the molecular structures of the cross-linker and network topological properties, such as end-to-end distance, crosslinking density and degree of conversion.     

Variational approximation methods for long-range force transmission in biopolymer gels

The variational principle of minimum free energy (MFEVP) has been widely used in research of soft matter statics. The MFEVP can be used not only to derive equilibrium equations (including both bulk equations and boundary conditions), but also to develop direct variational methods (such as Ritz method) to find approximate solutions to these equilibrium equations. We apply these variational methods to study long-range force transmission in nonlinear elastic biopolymer gels. It is shown that the slow decay of cell-induced displacements measured experimentally for fibroblast spheroids in three-dimensional fibrin gels can be well explained by variational approximations based on the three-chain model of biopolymer gels.     

PFPE脂润滑2Cr13钢摩擦副的真空摩擦磨损行为研究

利用销-盘式摩擦磨损试验机对PFPE脂润滑2Cr13钢摩擦副进行了不同滑动时间的真空滑动摩擦磨损试验.采用扫描电子显微镜(SEM)、X射线光电子能谱仪(XPS)和傅里叶变换红外光谱(FTIR)分别对磨损表面的形貌、化学状态及润滑脂的结构和成分进行了系统分析.结果表明:随着滑动时间的增加,由微切削磨损、轻微腐蚀磨损向严重腐蚀磨损、轻度局部剥落继续向严重黏着磨损、严重局部剥落的严重复合磨损转变.在真空摩擦过程中PFPE润滑脂主要以物理退化为主,即润滑脂中基础油和增稠剂的相对比例发生变化.同时,PFPE润滑脂与2Cr13钢在摩擦过程中发生化学反应,生成具有催化作用的Fe F3.但由于所生成的Fe F3量较少,并未使PFPE润滑脂发生化学降解.