2011年全国高中联赛一道题的三种解法

（2011年高中数学联赛第六题）在四面体ABCD中,已知∠ADB=∠BDC=∠CDA=60°,AD=BD=3,CD=2,则四面体ABCD的外接球的半径为.解法一由题意知：四面体的面△ABD是正三角形,又由余弦定理易知CA=CB=槡7.分别取AB,CD的中点K,M,取三角形△ABD中心N,连接CK,DK.在等腰三角形ABC中,     

一元一次不等式中的参数问题

我一直认为自己解不等式解得很好呢.但是今天我见了这样一道题:不等式组x2m-1无解,则m的取值范围是?我一下子晕了,这是我第一次碰到这样的问题:与以前的最大不同是有其他字母,后来我知道这个字母叫参数m,我多想把大于小于号换成等于号啊.没办法只好慢慢算了……终于我打算放     

Structural,electronic,vibrational,and thermodynamic properties of Zr_(1-x)Hf_xCo:A first-principles-based study

The physical properties including structural,electronic,vibrational and thermodynamic properties of Zr1-xHfx Co(x is the concentration of constituent element Hf,and changes from 0 to 1) are investigated in terms of the ABINIT program.The results reveal that all of Zr_(1-x)Hf_x Co have similar physical properties.When Hf concentration x gradually increases from 0.0 to 1.0,the lattice constant decreases from 3.217 ?A to 3.195 ?A very slowly.The calculated density of states(DOS)indicates that the metallic nature is enhanced and the electrical conductivity turns better with the increase of Hf.Moreover,as Hf concentration increases from 0 to 1,the Fermi energy gradually increases from-6.96 e V to-6.21 e V,and the electronic density of states at the Fermi level(N(E_f)) decreases from 2.795 electrons/e V f.u.down to 2.594 electrons/e V f.u.,both of which imply the decrease of chemical stability.The calculated vibrational properties show that the increase of Hf concentration from 0 to 1 causes the maximum vibrational frequency to decrease gradually from about 223 cm~(-1) to 186 cm~(-1),which suggests a lower dispersion gradient and lower phonon group velocities for these modes.Finally,the phonon related thermodynamic properties are obtained and discussed.     

Elastic,vibrational, and thermodynamic properties of Sr_(10)(PO_4)_6F_2 and Ca_(10)(PO_4)_6F_2 from first principles

The electronic, elastic, vibrational, and thermodynamic properties of Sr_(10)(PO_4)_6F_2(Sr-FAP) and Ca_(10)(PO_4)_6F_2(CaFAP) are systematically investigated by the first-principles calculations. The calculated electronic band structure indicates that the Sr-FAP and Ca-FAP are insulator materials with the indirect band gap of 5.273 eV and 5.592 eV, respectively. The elastic constants are obtained by the "stress–strain" method, and elastic modulus are further evaluated and discussed. The vibrational properties, including the phonon dispersion curves, the phonon density of states, the Born effective charge, and associated longitudinal optical and transverse optical(LO–TO) splitting of optical modes, as well as the phonon frequencies at zone-center are obtained within the linear-response approach. Substitution of Ca by Sr causes phonon frequencies to shift to lower values as expected due to the mass effect. Additionally, some phonon-related thermodynamic properties, such as Helmholtz free energy F, internal energy E, entropy S, and specific heat C_V of Sr-FAP and Ca-FAP are predicted with the harmonic approximation. The present calculated results of two apatites are consistent with the reported experimental and theoretical results.     

高阶思维是数学可视化教学的重要着力点

教师通过可视化技术开展数学教学是一种重要的教学方式,数学问题的动态可视可以提升教师"教"的效度,还可以帮助学生更好地认识数学知识,数学可视化带来了教学便利的同时,也引发思考:数学可视化教学的着力点是什么?本文以微专题教学案例的形式指出:高阶思维应是数学可视化教学的重要着力点.