On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Vision in click beetles (Coleoptera: Elateridae): pigments and spectral correspondence between visual sensitivity and species bioluminescence emission

Among lampyrids, intraspecific sexual communication is facilitated by spectral correspondence between visual sensitivity and bioluminescence emission from the single lantern in the tail. Could a similar strategy be utilized by the elaterids (click beetles), which have one ventral abdominal and two dorsal prothoracic lanterns? Spectral sensitivity [S(lambda)] and bioluminescence were investigated in four Brazilian click beetle species Fulgeochlizus bruchii, Pyrearinus termitilluminans, Pyrophorus punctatissimus and P. divergens, representing three genera. In addition, in situ microspectrophotometric absorption spectra were obtained for visual and screening pigments in P. punctatissimus and P. divergens species. In all species, the electroretinographic S(lambda) functions showed broad peaks in the green with a shoulder in the near-ultraviolet, suggesting the presence of short- and long-wavelength receptors in the compound eyes. The long-wavelength receptor in Pyrophorus species is mediated by a P540 rhodopsin in conjunction with a species-specific screening pigment. A correspondence was found between green to yellow bioluminescence emissions and its broad S(lambda) maximum in each of the four species. It is hypothesized that in elaterids, bioluminescence of the abdominal lantern is an optical signal for intraspecifc sexual communication, while the signals from the prothoracic lanterns serve to warn predators and may also provide illumination in flight.

Synthesis, spectral studies and X-ray crystal structure of N,N `-(+/-)-trans-1,2-cyclohexylenebis(3-ethoxysalicylideneamine) H-2(t-3-EtOsalchxn)

The synthesis, spectra and X-ray crystal structure of N,N`-(+/-)-trans-1,2-cyclohexylenebis(3-ethoxysalicylideneamine) H-2(t-3-EtOsalchxn), a salen-type ligand, are reported. The Schiff base was characterized by elemental analysis, m.p., IR, electronic spectra, H-1 and C-13 NMR spectra. The spectra are discussed and compared with those of N,N`-(+/-)-trans-1,2-cyclohexylenebis(salicylideneamine), H-2(t-salchxn). The electronic and IR spectra were also resolved by deconvolution. The influence of the ethoxy group on the IR, electronic spectrum, H-1 and C-13 NMR spectra is discussed. Strong intramolecular forces are present as supported by the IR and H-1 NMR spectra and the X-ray crystal structure. An intermolecular hydrogen bond is observed and appears twice in a pair of molecules in the unit cell. (c) 2007 Elsevier B.V. All rights reserved.