Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

Hybrid Electrodes Used in Polymer Electro-Optic Modulator with Ultra-Broadband

Hybrid electrode combined coplanar waveguide (CPW) and microstrip lines in ultra-broadband electro-optic modulators. The characteristics of the modulators with hybrid electrode is like that of the modulator with the microstrip lines, in which the microstrip electrode is loaded above only one arm of Mach-Zehnder (M-Z) optical waveguide, besides the problem of microstrip line to coaxial transition and corona polarization in polymer modulators have been best solved at the same time. By using finite element method (FEM), the characteristic parameters such as half-wave voltage, modulation bandwidth characteristic impedance, effective microwave refractive index of polymer modulators with microstrip line and hybrid electrode are analysed and compared in this paper. The results present that velocity match, impendence match and lower microwave propagating loss is easily realized in modulators with hybrid electrode.     

cos πρ theorems for δ-subharmonic functions

For ϕ a δ-subharmonic function, sharp results are obtained that connectA(r, ϕ), B(r, ϕ) andT(r, ϕ), whereA(r, ϕ)=inf_|z|=r ϕ(z),B(r, ϕ)=sup_|z|=r ϕ(z), andT(r, ϕ) is the Nevanlinna characteristics.     

Asymptotics of estimates in constrained nonlinear regression with long-range dependent innovations

The purpose of this paper is to investigate the asymptotic properties of the least squares estimates (L ₂-estimates) and the least absolute deviation estimates (L ₁-estimates) of the parameters of a nonlinear regression model subject to a set of equality and inequality restrictions, which has a long-range dependent stationary process as its stochastic errors. Then we will compare the asymptotic relative efficiencies of the above estimators.     

Nonextensive Statistical Mechanics Application to Vibrational Dynamics of Protein Folding

The vibrational dynamics of protein folding is analyzed in the framework of Tsallis statistics. We employ exact expressions for classical harmonic oscillators by considering the unnormalized constraints. As q→1, we show that these approximations agree with the result of Gaussian network model.     

Evidences of phase separation in moisture‐cured poly(urethane urea)s by means of temperature modulated differential scanning calorimetry

The degree of phase separation in several moisture‐cured poly(urethane urea)s (PUUs) was studied by FTIR spectroscopy, wide angle X‐ray diffraction (WAXD), and temperature‐modulated differential scanning calorimetry (TMDSC). This latter technique was shown to be particularly useful in analysing the degree of phase separation in PUU polymers. Both phase mixing and phase segregation coexisted in the PUUs and the degree of phase separation increased as the urea hard segment (HS) content in the PUU increased. The maximum solubility of urea HSs into the polyol soft segments (SSs) was achieved for 50 wt % urea HS content in diol‐based PUUs, whereas for triol‐based PUUs the highest solubility between HS and SS was reached for lower urea HS amount. Finally, the higher the urea HS content the higher the extent of phase separation in the PUU. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3034–3045, 2007     

环境对MgB_2氧化过程中MgO晶须生长形状的影响

用扫描电镜研究了在 Mg B2 氧化过程中环境对 Mg O晶须生长形状的影响。仅有 Mg B2 时 ,在 80 0℃左右空气气氛中 ,可生长出纳米尺寸的 Mg O晶须。若把 Mg B2 添加到氧化铝陶瓷基体中 ,则生长成竹节状晶须。Mg O晶须生长形状的差别可能是由于氧浓度以及位错等因素造成的。