A theoretical experiment using gravitational optics to find the density function of a star

The gravitational deflection of null geodesics passing through a static, spherically symmetric star, as a function of impact parameter, is related to the energy density, as a function of radius. We exhibit this deflection function graphically for two solar models and for Chandrasekhar white dwarfs, assuming small angles, linear superposition of deflections, and negligible pressure. The caustic and axicon infinities in the intensity of short-wave penetrating radiation—neutrinos or very short gravity waves—are described. It is argued that experimental knowledge of the axicon intensity function and of the real caustic locus uniquely define the stellar density function, in the usual case of one virtual caustic branch. The two solar models give rather different distances from the sun to the paraxial focus: 21 and 30 au, unfortunately the nearest infinity. Transverse misalignment of only 7 m, for the case of 21 au, reduces the infinite gain at the paraxial focus to 10⁹.     

An approach to the numerical solutions of geometric programs

An algorithm for solving ordinary geometric programs is presented. The algorithm is based on the reduced system associated with geometric programs and is highly flexible in that it allows the use of several nonlinear optimization techniques.     

Some considerations of the steady-state and transient behaviour of membrane-covered dissolved oxygen detectors

The behaviour of commercial, membrane-covered dissolved oxygen detectors is considerably more complicated than the orginal model of Mancy et al. would suggest. For example, the steady-state current is dependent both on the thickness and nature of the electrolyte behind the membrane. And transient currents obtained for switch-on of the detector are not completely described by the simple diffusional model. This paper examines these examples of more complicated behaviour. An analysis of the transient current for a step change in dissolved oxygen concentration is also given.     

Some congruence properties of three well-known sequences: Two notes

The three sequences mentioned in the title are Ramanujan's τ-function, the coefficients c_n of Klein, Fricke, and Shimura, and the sequence a_n of Apèry numbers. In the first note, it is shown that c_n ≡ τ(n)(mod 11). In the second note it is shown that for a prime p, a_p+1 ≡ 25 + 60p(mod p²).     

Plane magnetohydrodynamic flows with constantly inclined magnetic and velocity fields

Summary The general solution is given for steady two-dimensional magnetohydrodynamic flows of an incompressible, inviscid, fluid having infinite electrical conductivity when the angle between the magnetic field and the velocity field is every where constant.

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Résumé On donne ici la série complète des solutions pour les écoulements magnéto-hydrodynamiques permanents bidimensionnels d'un fluide incompressible, non-visqueux ayant une conductivité électrique infinie lorsque l'angle entre le champ magnétique et le champ de vitesse est partout constant.

     

Stochastic equations of motion with damping

A nonlocal equation of motion with damping is derived by means of a Mori-Zwanzig renormalization process. The treatment is analogous to that of Mori in deriving the Langevin equation. For the case of electrodynamics, a local approximation yields the Lorentz equation; a relativistic generalization gives the Lorentz-Dirac equation. No self-acceleration or self-mass difficulties occur in the classical treatment, although runaway solutions are not eliminated. The nonrelativistic quantum case does not exhibit runaways, however, provided one remains within a weak damping approximation. The correspondence limit shows that a classical limit may be taken, again within the same approximation.