The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction A_B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces and such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible. James Hinchcliffe and Serguei Naboko wish to thank the British EPSRC for financial support under grant EP/C008324/1 “Spectral Problems on Families of Domains and Operator M-functions”. Serguei Naboko wishes to thank the Russian RFBR for grant 06-01-00219. All authors wish to thank INTAS for financial support under INTAS Project No. 051000008-7883. The authors wish to thank the referee for many helpful comments.     

Electronic structure of diamagnetic and paramagnetic hexanuclear chalcohalide clusters of rhenium

Hexanuclear chalcohalide clusters of rhenium(III) of general formula [Re(6)S(4+x)Cl(10-x)](x-) with x = 1-4 have been studied using quantum chemical DFT calculations. The optimized structures reproduce the geometrical features found from X-ray data for the members of the series. The relative stability of different stereoisomers for the mono- and dianions has been estimated. The analysis of the tetraanion series [Re(6)Q(8)X(6)](4-) with Q = S, Se and X = Cl, Br, I, and CN demonstrates the influence of the mu(1)- and mu( 3)-ligands on the strength of Re-apical ligand bond. It is shown that the tetragonal distortion found for the stable oxidized paramagnetic species [Re(6)S(8)Cl(6)]*(3-) results from the Jahn-Teller effect for a doubly degenerate electronic state.     

Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

We study distributions F on [0,) such that for some T , F ^*2(x, x+T] 2F(x, x+T]. The case T = corresponds to F being subexponential, and our analysis shows that the properties for T < are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.     

Generalized fractal dimensions: equivalences and basic properties

Given a positive probability Borel measure μ on , we establish some basic properties of the associated functions τ_μ^±(q) and of the generalized fractal dimensions D_μ^±(q) for . We first give the equivalence of the Hentschel–Procaccia dimensions with the Rényi dimensions and the mean-q dimensions, for q>0. We then use these relations to prove some regularity properties for τ_μ^±(q) and D_μ^±(q); we also provide some estimates for these functions, in particular estimates on their behaviour at ±∞, as well as for the dimensions corresponding to convolution of two measures. We finally present some calculations for specific examples illustrating the different cases met in the article.     

Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lüst operator

We consider a matrix differential operator with singular entries which arises in magnetohydrodynamics. By means of the asymptotic Hain-Lüst operator and some pseudo-differential operator techniques, we determine the essential spectrum of this operator. Whereas in the regular case, the essential spectrum consists of two intervals, it turns out that in the singular case two additional intervals due to the singularity may arise. In addition, we establish criteria for the essential spectrum to lie in the left half-plane.

     

Lysine-enediyne conjugates as photochemically triggered DNA double-strand cleavage agents

Statistical analysis of DNA-photocleavage by two types of lysine-enediyne conjugates confirms that more double-strand breaks are produced than can be accounted for by coincident single-strand breaks.     

Symmetric kernel of Rademacher multiplicator spaces

Let X be a rearrangement invariant (r.i.) function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of measurable functions x such that xh∈X for every a.e. converging series h=∑a_nr_n∈X, where (r_n) are the Rademacher functions. We show that for a broad class of r.i. spaces X, the space Λ(R,X) is not r.i. In this case, we identify the symmetric kernel of the Rademacher multiplicator space and study when reduces to L^∞. In the opposite direction, we find new examples of r.i. spaces for which Λ(R,X) is r.i. We consider in detail the case when X is a Marcinkiewicz or an exponential Orlicz space.     

Observation of spatial migration of excitation in Er-doped optical fibers by means of a population grating technique

The influence of spatial migration of excitation among neighboring Er3+ ions on the dynamics of population grating formation in Er-doped optical fibers is reported. The effect manifests itself in an additional increment in the grating formation rate compared with the fluorescence growth rate observed for the same spatially uniform average light power. The experiments, performed at lambda=1549 nm in the configuration of transient two-wave mixing with two similar single-mode Er-doped fibers with significantly different erbium concentrations (approximately 640 and approximately 5600 parts in 10(6)), demonstrated the essential contribution of this effect to the grating formation rate in the latter fiber. The evaluated diffusion coefficient proved to be approximately 2.3 x 10(-9) cm2/s, which ensures effective migration of the excitation by approximately 48 nm, i.e., by approximately 18 average inter-ion distances in this fiber.     

Ab initio study of a biradical radiationless decay channel of the lowest excited electronic state of cytosine and its derivatives

A theoretical model for the ultrafast S1-->S0 internal conversion of cytosine is presented, in which a state switch from the initially prepared 1pipi* state to the out-of-plane deformed excited state of biradical character controls the rate of the S1(1pipi*) decay. This mechanism successfully accounts for the dramatically longer S1 lifetimes of 5-fluorocytosine and N-acetylcytosine relative to cytosine. The replacement of the C5 hydrogen atom by a methyl group is predicted to lead to a substantial, but not dramatic, increase in the S1 lifetime, also consistent with experiment. It is this ability to correctly predict the substituent effects that distinguishes the present model from the previously proposed mechanisms.     

Electron Localization versus Delocalization and Ferromagnetic Coupling in Mixed Valence Polyoxometalates

We show that the ferromagnetism, recently found in the doubly-reduced hexanuclear polyoxometalate [V₆O₁₂ (μ₂−OCH₃)₄(L)₄] does not contradict to the proposed earlier delocalization mechanism of the stabilization of a spin- singlet state. The ground state triplet state yields from the internal and/or external complex asymmetry leading to the spin density localization.