Eigenvalues of the Laplacian acting on -forms and metric conformal deformations

Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

Laser safety: The user's problems and requirements for the solutions

The findings of an interview programme carried out with a representative sample of the UK laser community in order to establish their problems and requirements with regard to laser safety is reported. The sample comprised laser manufacturers, laser users and inspectors of the UK Health and Safety Executive (HSE). The results indicated that 80% of interviewees had experienced problems in using the current UK Laser Safety Standards. The usefulness of these Standards is reduced when the broader laser user community is considered. Analysis of user requirements highlighted the importance of laser application-orientated laser safety material and training. Some areas of laser safety were identified as requiring further investigation.     

Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace–Beltrami operator on closed surfaces of unit area.     

Host‐Assisted Guest Self‐Assembly: Enhancement of the Dimerization of Pyronines Y and B by γ‐Cyclodextrin

Buckle up! The dimerization of small fluorescent guests is strongly enhanced in presence of a cyclodextrin host. The host cavity acts like a belt to assist the self‐assembly of guests (see picture). Small variations in the guest structure have significant influence on the stability and geometry of the aggregates.

     

Author Index, Volume 24 (2003)

Authors Index

Author Index, Volume 24 (2003)     

Fluorescence‐Labeled Bis‐benzamidines as Fluorogenic DNA Minor‐Groove Binders: Photophysics and Binding Dynamics

In recent decades there has been great interest in the design of highly sensitive sequence‐specific DNA binders. The eligibility of the binder depends on the magnitude of the fluorescence increase upon binding, related to its photophysics, and on its affinity and specificity, which is, in turn, determined by the dynamics of the binding process. Therefore, progress in the design of DNA binders requires both thorough photophysical studies and precise determination of the association and dissociation rate constants involved. We have studied two bis‐benzamidine (BBA) derivatives labeled by linkers of various lengths with the dye Oregon Green (OG). These fluorogenic binders show a dramatic fluorescence enhancement upon binding to the minor groove of double‐stranded (ds) DNA, as well as significant improvement in their sequence specificity versus the parent BBA, although with decreased affinity constants. Detailed photophysical analysis shows that static and dynamic quenching of the OG fluorescence by BBA through photoinduced electron transfer is suppressed upon insertion of BBA into the minor groove of DNA. Fluorescence correlation spectroscopy yields precise dynamic rate constants that prove that the association process of these fluorogenic binders to dsDNA is very similar to that of BBA alone and that their lower affinity is mainly a consequence of their weaker attachment to the minor groove and the resultant faster dissociation process. The conclusions of this study will allow us to go one step further in the design of new DNA binders with tunable fluorescence and binding properties.