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.     

Dynamic response of doubly curved honeycomb sandwich panels to random acoustic excitation. Part 2: Theoretical study

In this paper a single-degree-of-freedom model is developed to predict the dynamic response of an acoustially excited doubly curved sandwich panel. Three variants of the model are investigated, based on differing assumptions regarding the spatial distribution of the applied loading. When the loading is assumed to be uniform then the model reduces to the Miles approach, and when the loading is assumed to conform to the structural mode shape then the method is very similar to the Blevins approach. The third variant involves a more detailed consideration of the travelling wave characteristics of the applied loading, and this is found to give much improved agreement with experimental results obtained in a progressive wave tube facility. In addition, an approach using the finite element method is presented in which the response to grazing incidence excitation is computed, and this is also found to yield good agreement with the experimental results.     

On the Frequency of Zeros of Solutions of Second Order Linear Differential Equations

We consider the equation \(\rm f^{\prime\prime}+{A}(z){f}=0\) with linearly independent solutions f₁,₂, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f₁),λ(f₂)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of \(\rm f^{\prime\prime}+{e}^P{f}=0\) satisfies λ(f) = ∞. Finally we consider the particular equation \(\rm f^{\prime\prime}+({e}^Z-K){f}=0\) where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f₁, f₂ each with λ(f_i) ≤ 1.     

The zeros of the first two derivatives of a meromorphic function

We prove a theorem which implies the following: if is meromorphic of finite order in the plane and and have only finitely many zeros, then has only finitely many poles.

     

The design of an irradiator for the continuous processing of liquid latex

This paper presents anew design concept for a gamma irradiation plant for the continuous processing of pumpable liquids. Typical applications of such a plant include

• * the irradiation vulcanisation of natural latex rubber

• * disinfection of municipal sewage sludge for agricultural use

• * sterilisation of liquids in the pharmaceutical and cosmetics industries

• * industrial processing of bulk liquids

The authors describe the design and operation of the latex irradiator now operating on a small production scale in Malaysia and proposed developments.

The design allows irradiation processing to be carried out under an inert or other gaseous environment.

State-of-the-art computer control system ensures the fully automatic processing operation needed by industrial computers.     

Selective interactions of cationic porphyrins with G-quadruplex structures

G-quadruplex DNA presents a potential target for the design and development of novel anticancer drugs. Because G-quadruplex DNA exhibits structural polymorphism, different G-quadruplex typologies may be associated with different cellular processes. Therefore, to achieve therapeutic selectivity using G-quadruplexes as targets for drug design, it will be necessary to differentiate between different types of G-quadruplexes using G-quadruplex-interactive agents. In this study, we compare the interactions of three cationic porphyrins, TMPyP2, TMPyP3, and TMPyP4, with parallel and antiparallel types of G-quadruplexes using gel mobility shift experiments and a helicase assay. Gel mobility shift experiments indicate that TMPyP3 specifically promotes the formation of parallel G-quadruplex structures. A G-quadruplex helicase unwinding assay reveals that the three porphyrins vary dramatically in their abilities to prevent the unwinding of both the parallel tetrameric G-quadruplex and the antiparallel hairpin dimer G-quadruplex DNA by yeast Sgs1 helicase (Sgs1p). For the parallel G-quadruplex, TMPyP3 has the strongest inhibitory effect on Sgs1p, followed by TMPyP4, but the reverse is true for the antiparallel G-quadruplex. TMPyP2 does not appear to have any effect on the helicase-catalyzed unwinding of either type of G-quadruplex. Photocleavage experiments were carried out to investigate the binding modes of all three porphyrins with parallel G-quadruplexes. The results reveal that TMPyP3 and TMPyP4 appear to bind to parallel G-quadruplex structures through external stacking at the ends rather than through intercalation between the G-tetrads. Since intercalation between G-tetrads has been previously proposed as an alternative binding mode for TMPyP4 to G-quadruplexes, this mode of binding, versus that determined by a photocleavage assay described here (external stacking), was subjected to molecular dynamics calculations to identify the relative stabilities of the complexes and the factors that contribute to these differences. The DeltaG(o) for the external binding mode was found to be driven by DeltaH(o) with a small unfavorable TDeltaS(o) term. The DeltaG(o) for the intercalation binding model was driven by a large TDeltaS(o) term and complemented by a small DeltaH(o) term. One of the main stabilizing components of the external binding model is the energy of solvation, which favors the external model over the intercalation model by -67.94 kcal/mol. Finally, we propose that intercalative binding, although less favored than external binding, may occur, but because of the nature of the intercalative binding, it is invisible to the photocleavage assay. This study provides the first experimental insight into how selectivity might be achieved for different G-quadruplexes by using structural variants within a single group of G-quadruplex-interactive drugs.