Theory of Valuations on Manifolds: A Survey

This is a non-technical survey of a recent theory of valuations on manifolds constructed in [A10], [A11], [AF] and [A12], and actually a guide to this series of articles. We also review some recent related results obtained by a number of people. Received: February 2006, Revision: June 2006, Accepted: June 2006     

Towards a new edition of the “Guide to the expression of uncertainty in measurement”

The “Guide to the expression of uncertainty in measurement” (GUM) is an extremely important document. It unifies methods for calculating measurement uncertainty and enables the consistent interpretation and comparison of measurement results, regardless of who obtained these measurements and where they were obtained. Since the document was published in 1995, it has been realised that its recommendations do not properly address an important class of measurements, namely, non-linear indirect measurements. This drawback prompted the initiation of the revision of the GUM in the Working Group 1 of the Joint Committee for Guides in Metrology, which commenced in October 2006. The upcoming revision of the GUM provides the metrological community with an opportunity to improve this important document, in particular, to reflect developments in metrology that have occurred since the first GUM publication in 1995. Thus, a discussion of the directions for this revision is important and timely. By identifying several shortcomings of the GUM and proposing directions for its improvement, we hope this article will contribute to this discussion. Papers published in this section do not necessarily reflect the opinion of the Editors, the Editorial Board and the Publisher.     

Characteristic functions of almost solvable extensions of Hermitian operators

The characteristic operator-functions W() are studied of the almost solvable extensions of an Hermitian operator. The inverse problem is solved, a multiplication theorem is proved, and a formula is derived expressing W() in terms of the Weyl function and the boundary operator. Characteristic functions are computed of various differential and difference operators, with the help of which are proved theorems of the completeness of the systems of proper and adjoint vectors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 435–459, April, 1992.     

Simulations of radiative effects on the Rayleigh–Taylor instability using the CRASH code

Future experiments at the National Ignition Facility will be able to generate diagnosable Rayleigh–Taylor instability growth in the presence of locally generated, high radiation fluxes. This interplay of radiative energy transfer and hydrodynamic instability is relevant to many astrophysical systems, such as core-collapse red supergiant supernovae. Previous simulations of high-energy-density Rayleigh–Taylor instabilities in the presence of a hot environment near a radiative shock demonstrate behavior that differs from that found in non-radiative cases. However, these simulations considered only 1D or single wavelength cases. Here we report simulations of an entire experimental system using the CRASH code. These simulations lead to modified predictions, attributed to the effects of radial energy losses.     

On the Deficiency Indices of Block Jacobi Matrices Related to Dirac Operators with Point Interactions

Mathematical Notes -     

Trace formulas for additive and non-additive perturbations

Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H^′,H}$\{H^{\prime },H\}$ of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H   and H^′=H+V$H^{\prime }=H+V$ are maximal accumulative and V is trace class, we prove the existence of a summable   complex-valued spectral shift function. We also obtain trace formulas for pairs {H,H^?}$\{H,H^{?}\}$assuming only that H and  H^?$H^{?}$are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas.     

Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole

We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of \mathbb C{\mathbb C} . As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed.     

Nonnegative Fourier–Jacobi Coefficients and Some Classes of Functions

Fourier–Jacobi series with nonnegative Fourier–Jacobi coefficients are considered. Under special restrictions on the Jacobi weight function, we establish in terms of Fourier–Jacobi coefficients a necessary and sufficient condition in order that the sum of the Fourier–Jacobi series should possess certain structural properties.