Almost harmonic spinors

We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.     

Diazadioxa[8]circulenes: Planar Antiaromatic Cyclooctatetraenes

In this paper we describe a new class of antiaromatic planar cyclooctatetraenes: the diazadioxa[8]circulenes. The synthesis was achieved by means of a new acid‐mediated oxidative dimerization of 3,6‐dihydroxycarbazoles to yield the diazadioxa[8]circulenes in high yields. The synthetic protocol appears to be general, and is a one‐pot transformation in which two C? C bonds and two C? O bonds are formed with the loss of two molecules of water. We also present a detailed characterization of the optical and electrochemical properties of this new class of stable planar cyclooctatetraenes. The properties of the diazadioxa[8]circulenes are compared with the properties of isoelectronic tetraoxa[8]circulenes and azatrioxa[8]circulenes. We discuss the antiaromatic nature of the planar central cyclooctatetraene moiety. The antiaromatic nature of the planar cyclooctatetraenes was studied by using computational methods (NICS calculations), and these calculations reveal that the central eight‐membered ring has antiaromatic character.     

Nanoring as a magnetic or electric field sensitive nano-antenna for near-field optics applications

In this paper we investigate, in the visible range, the behavior of the electromagnetic near-field, perturbed by a metal nanoring. The vectorial study which is based on FDTD simulation, consists of comparing in the near zone, the electric and magnetic field distributions while interacting with the metal nanoring, according to the incident polarization state. Both image distributions and chromatic spectra are computed and analyzed. The purpose of this work is the enhancement of either the electric or the magnetic emission/detection capability of nanoring according to illumination conditions. Such nano-objects can be seen as particular selective nanodevices playing the role of nano-antennas usable in near-field microscopy as an alternative solution to usual tips.     

All-fiber controller of radial polarization using a periodic stress

Our aim is to transpose the polarization control by mechanical stress, usually applied to single-mode fibers, to the (TM(01), TE(01), HE(21)(ev), HE(21)(od)) annular mode family. Nevertheless, the quasi-degeneracy of these four modes makes the situation more complex than with the fundamental mode HE(11). We propose a simple device based on periodic perturbation and mode coupling to produce the radially polarized TM(01) mode or at least one of the four modes at the extremity of an arbitrarily long fiber, the conversion to TM(01) mode being achievable by classical crystalline plates.     

A simple Hill-series approach to the linear potential

A simple Hill-series method is shown to yield accurate energy levels and expectation values for the linear potential for any angular momentum L and confinement radius R. The numerical results are verified by comparison with those from two hypervirial perturbation methods which are specially constructed to give accurate results at L = 0 for small R values and at L >?1 for R =?∞.     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

The orthogonality property of the Lommel polynomials and a twofold infinity of relations between Rayleigh''s σ-sums

Making use of a remarkable theorem which expresses a relationship between a certain type of infinite continued fractions and systems of orthogonal polynomials, it is proven that the known infinite continued fraction development of the ratio of Bessel functions J_v?1(z)/J_v(z) gives rise to an orthogonality property of the Lommel polynomials $\text{{R}{}_{\mathrm{m,v}}\text{(}\text{1}\text{z}\text{)|m?}\text{N}\text{}}$ when v is real and positive. The corresponding weight function which appears to be non-negative in the interval of definition, is obtained by the application of two successive integral transforms. It consists of an infinite series of Dirac δ-functions whose singularities are distributed symmetrically around the origin on the real axis in such a manner that the origin is their limit point on both sides. For any positive v, the Lommel polynomials form a system of so-called orthogonal polynomials of a discrete variable. The orthogonality property may also be conveniently expressed by means of a Stieltjes integral. One of its corollaries is a twofold infinity of linear relations between the sums σ_v^(r) defined by σ_v^(r)=Σ_n=1^+∞1/j_v,n^2r, with $\text{v+1Σ}\text{R}{}_0{}^{\mathrm{+}}\text{, rΣ}\text{N}{}_0\text{,}\text{in which}\text{j}{}_{\mathrm{v,n}}\text{represents the}\text{n}\text{th positive zero of}\text{J}{}_{\mathrm{v}}\text{(z)}$.Another by-product consists of a complement to a theorem of Hurwitz concerning the nature and the position of the zeros of the Lommel polynomials written as g_m,v(z) in the modified notation of the mentioned author. From this study also result two interesting approximations of j_v,1 applicable for v?]?1, +1].     

Alkylation des «malonyl-α-aminopyridines»

The alkylation of the ‘malonyl-α-aminopyridines’ ( 4 ) gives mixtures of substitution products, mainly the N¹-and O²-alkyl derivatives. The IR., UV. and NMR. spectra are discussed.     

Smallest lithographic marks generated by optical focusing systems

We show that the combination of Bessel beams and photosensitive materials exhibiting polarization filtering properties allows one to reach the smallest mark that can be lithographically generated by focusing systems. This property is of interest in current optical data storage techniques.