从2006年高考看用导数探讨函数图象的交点问题

2006年高考数学导数命题在方向基本没变的基础上，又有所创新．导数命题创新的两个方面：一是研究对象的多元化，由研究单一函数转向研究两个函数或多个函数，二是研究内容的多元化，由用导数研究函数性质（单调性、最值、极值）转向运用导数综合研究函数的性质、函数图象的交点和方程根的分布等，实际上就是运用导数考查函数图象的交点个数问题．     

ON A LEMMA OF DIPERNA AND CHEN

This note gives an explicit lower bound of the function (?)(t) in the lemma of DiPerna and Chen and some remarks.     

Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.     

柠檬酸在烟草上的吸附性能研究

采用静态法，研究了柠檬酸在烟草上的吸附和解吸附性能以及共存物质对吸附能力的影响，测定了在不同浓度、温度下的吸附速率曲线和吸附等温线，并估算出吸附热。结果表明，柠檬酸在烟草上的吸附量大，吸附速率快，属物理吸附。当柠檬酸的浓度小于0．005mol/L时，吸附等温线呈直线型；而当柠檬酸的浓度增大到0．15mol/L的范围内时，吸附等温线则呈现典型的“台阶”型。     

对范德瓦耳斯等温线上存在的亚稳性质的讨论

本文用热力学理论，论证了范氏等温线上某些态的亚稳性质。     

表面活性剂在固/液界面吸附理论的新进展

根据两阶段吸附模型~[1], 导出了两常数及一常数吸附等温线公式。不仅可以方便地应用于各种类型的表面活性剂在固/液界面上的吸附数据, 而且具有预示能力。对于显示LS型等温线的体系只需测定第一平台处和饱和的吸附量即可给出整个的吸附等温线。对于离子型表面活性剂在反电性固体上的吸附, 可自一个化合物的吸附预示出一系列同系物在同一吸附剂上的吸附等温线。