On a Fermat principle in general relativity. A Ljusternik-Schnirelmann theory for light rays

Summary In this paper existence and multiplicity results for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions. Such results are obtained using an extension to Lorentzian Geometry of the classical Fermat principle in optics. The results are proved using critical point theory on infinite dimensional manifolds. An application to the gravitational lens effect is presented.     

A timelike extension of Fermat's principle in general relativity and applications

We define a variational problem based on the arrival time functional for timelike curves on a Lorentzian manifold parameterized by a fixed constant multiple of their proper time. Under a causality assumption for the manifold , we prove that the stationary points of our problem are geodesics, obtaining an extension of the Fermat's Principle for light rays proven in [14] (see also [2]). Moreover, we study the compactness pr operties of the arrival time functional by global variational techniques. Under intrinsic assumptions on the metric of we get results of existence and multiplicity for geodesics with a given energy between an event and an observer of .     

Geodesics on lorentzian manifolds with quasi-convex boundary

In this paper we study existence and multiplicity results of geodesics joining two given events in Lorentzian manifolds with lack of geodesic completeness. The considered Lorentzian manifolds are not necessarily static or stationary and satisfy a condition of convexity on the boundary. work supported by M.U.R.S.T. research founds 40%–60% 1992     

Multiple ionization of argon in coincidence with projectile ions in 60–120 MeV Si q+-Ar collisions

A projectile ion-recoil ion coincidence technique has been employed to study the multiple ionization and the charge transfer processes in collisions of 60–120 MeV Si ^q+ (q = 4−14) ions with neutral argon atoms. The relative contribution of different ionization channels, namely; direct ionization, electron capture and electron loss leading to the production of slow moving multiply charged argon recoil ions have been investigated. The data reported on the present collision system result from a direct measurement in the considered impact energy for the first time. The total ionization cross-sections for the recoil ions are shown to scale as q ^1.7/E _p ^0.5 , where E _p is the energy in MeV of the projectile and q its charge state. The recoil fractions for the cases of total- and direct ionizations are found to decrease with increasing recoil charge state j. The total ionization fractions of the recoils are seen to depend on q and to show the presence of a ‘shell-effect’ of the target. Further, the fractions are found to vary as 1/j ² upto j = 8+. The average recoil charge state 〈j〉 increases slowly with q and with the number of lost or captured electrons from or into the projectile respectively. The projectile charge changing cross-sections σ _qq′ are found to decrease with increasing q for loss ionization and to increase with q for direct-and capture ionization processes respectively. The physics behind various scaling rules that are found to follow our data for different ionization processes is reviewed and discussed.     

Genericity of Nondegeneracy for Light Rays in Stationary Spacetimes

Given a Lorentzian manifold (M, g), an event p and an observer U in M, then p and U are light conjugate if there exists a lightlike geodesic γ : [0, 1] → M joining p and U whose endpoints are conjugate along γ. Using functional analytical techniques, we prove that if one fixes p and U in a differentiable manifold M, then the set of stationary Lorentzian metrics in M for which p and U are not light conjugate is generic in a strong sense. The result is obtained by reduction to a Finsler geodesic problem via a second order Fermat principle for light rays, and using a transversality argument in an infinite dimensional Banach manifold setup.     

Overcoming the boundary effects in surface spline interpolation

Surface spline interpolation when the domain is all of R^d isknown to converge much faster to the data function f than inthe case when the domain is the unit ball. This difference isunderstood to be due to boundary effects which, as will be shown,also affect the size of the surface spline's coefficients. Wepropose a modified form of surface spline interpolation which,to a great extent, overcomes these boundary effects. This modifiedsurface spline interpolant uses only the values of f at thegiven interpolation points.     

Multiplicity of solutions for elliptic inequalities in presence of symmetries

Summary In this paper we get a multiplicity result for the solutions of a variational inequality of elliptic type, in presence of symmetries.

---

Riassunto Viene fornito un teorema di molteplicità di soluzioni per una disequazione variazionale di tipo ellittico in presenza di simmetrie.

---

---

Work supported by M.P.I. and G.N.A.F.A.     

Carbon-13 EPR Hyperfine Interaction in the Complex [Rh(CN)6]in a KCl Host Lattice

The hyperfine interaction with¹³C in nonenriched [Rh(CN)₆]⁴⁻, in a KCl host lattice at 7 K, is measured by continuous-wave X- and Q-band electron paramagnetic resonance, allowing, for the first time, a comparative study of spin densities on the carbon of the axial cyanide in this complex and in the similar [Co(CN)₆]⁴⁻complex. From the experimental data corrected for dipolar interactions,A_iso= 79.80 × 10⁻⁴cm⁻¹andA_anis= 0.76 × 10⁻⁴cm⁻¹.