Stable hysteretic reflection of light at a nonlinear interface

We have observed a hysteretic (bistable) reflection of a Gaussian laser beam at the interface between glass and a liquid solution of polystyrene microspheres as nonlinear medium with both branches of the hysteretic curve being stable, in contrast to the results of previous experiments.     

On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trindX?α. W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trIndX?λ. T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dimWX?α where dimW is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dimC of the covering dimension.As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dimCX?α belonging to some absolute Borel class.     

One cannot hear orbifold isotropy type

We show that the isotropy types of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. Indeed, we construct arbitrarily large families of mutually isospectral orbifolds with different isotropy types. Finally, we show that the corresponding singular strata of two isospectral orbifolds may not be homeomorphic. Received: 6 October 2005     

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Determinants of Laplacians in real line bundles over hyperbolic manifolds connected with quantum geometry of membranes

The possibility is discussed of generalizing the Polyakov approach to strings on membranes and the connection of such a generalization with Thurston's classification of three-dimensional geometries. The important ingredients for computing a membrane path integral are the determinants of scalar Laplacians acting in real line bundles over three-dimensional closed manifolds. In the closed bosonic membrane case, such determinants are evaluated for a class of closed 3-manifolds of the H ³/ form with a discrete subgroup of isometries of the three-dimensional Lobachevsky space H ³ and they are expressed in terms of the Selberg zeta function. Some further possible implications of the results obtained are also discussed.     

Hölder exponents and box dimension for self-affine fractal functions

We consider some self-affine fractal functions previously studied by Barnsleyet al. The graphs of these functions are invariant under certain affine scalings, and we extend their definition to allow the use of nonlinear scalings. The Hölder exponent,h, for these fractal functions is calculated and we show that there is a larger Hölder exponent,h , defined at almost every point (with respect to Lebesgue measure). For a class of such functions defined using linear affinities these exponents are related to the box dimensionD _B of the graph byh2–D _Bh .Communicated by Michael F. Barnsley.     

Heat “polynomials” for a singular differential operator on (0, ∞)

We generalize the theory of the heat polynomials introduced by P. V. Rosenbloom and D. V. Widder for a more general class of singular differential operator on (0, ). The heat polynomials associated with the Bessel operator and studied by D. T. Haimo appear as a particular case in this paper. In the special cases of second derivative and Bessel operators the heat polynomials are in fact polynomials inx andt, however, this property does not hold in general.Communicated by Tom. H. Koornwinder.     

Algorithms for non-linear huber estimation

The Huber criterion for data fitting is a combination of thel ₁ and thel ₂ criteria which is robust in the sense that the influence of wild data points can be reduced. We present a trust region and a Marquardt algorithm for Huber estimation in the case where the functions used in the fit are non-linear. It is demonstrated that the algorithms converge under the usual conditions.     

Symmetries and conservation laws of Kadomtsev-Pogutse equations

Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, enable us to demonstrate their potentials in action. This paper presents, firstly, the results of computations of symmetries and conservation laws for these equations and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by the considered equations are exhausted by those enumerated in this paper. Secondly, we point out some reductions of Kadomtsev-Pogutse equations to more simpler forms which have less independent variables and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and their physical interpretation are demonstrated.