Some Isoperimetric Inequalities in Electrochemistry and Hele Shaw Flows

Isoperimetric inequalities are applied to a moving-boundaryproblem for doubly-connected domains. This problem occurs forexample in electrochemistry, in which case the domains in questionare the electrolyte of an electrolytic cell. The two electrodessurrounding the electrolyte are assumed to grow or dissolve,at different rates in general, by electrochemical reaction.We obtain optimal estimates showing, for example, that the leastchange in volume of each electrode always occurs in sphericalsymmetry.     

Sequences of Diophantine approximations

We sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of “good” Diophantine approximations to a fixed α ∈ C are trivial ones. For example, suppose that a > 1 and that (δ_n)_n=1^∞ and (σ_n)_n=1^∞ are two positive, strictly increasing unbounded sequences satisfying δ_n+1 ≤ aδ_n and σ_n+1 ≤ aσ_n. If there is a sequence of nonzero polynomials P_n ∈ Z[x] with deg P_n ≤ δ_n, deg P_n + log height P_n ≤ σ_n, and ∣P_n(α)∣ ≤ e^{?(2a+1)δ_nσ_n}, then each P_n(α) = 0.     

The algebraic independence of certain numbers related by the exponential function

We give a variation of a theorem of Gelfond. One of the corollaries is Schneider's conjecture that either exp(e) or exp(e²) is transcendental. It also follows that at least one of the numbers exp(α), exp(α²), exp(α³) is transcendental for any nonzero complex number α.     

Bi2223带材临界电流沿长度方向的统计分布

本文采用四引线法测量了Bi2223带材临界电流沿长度方向的分布,采用正态分布、对数正态分布、威布尔分布和最小极值分布对临界电流分布进行了拟合检验,确定了临界电流的最优统计分布模型.