The first main theorem on complements: from global to local

The aim of this paper is to clarify and generalize techniques of works alg-geom/9711024 (see also math.AG/9810097 and math.AG/9901004). Roughly speaking, we prove that for local Fano contractions the existence of complements can be reduced to the existence of complements for lower dimensional projective Fano varieties.Comment: 27 pages, LaTeX2

On birational involutions of P3P^3

Let $X$ be a rationally connected three-dimensional algebraic variety and let $\tau$ be an element of order two in the group of its birational selfmaps. Suppose that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program we give a rough classification of such elements.Comment: 24 pages, late

Approximation with Random Bases: Pro et Contra

In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both randomized and deterministic, for selecting elements from these families that have been shown to ensure the rate of convergence in $L_2$ norm of order $O(1/N)$, where $N$ is the number of elements. We show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided. In the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and/or extreme sensitivity of the outcome of the approximation to parameters. Implications of our analysis for applications of neural networks in modeling and control are illustrated with examples.Comment: arXiv admin note: text overlap with arXiv:0905.067

ROSAT X-ray sources and exponential field decay in isolated neutron stars

In this paper we semianalyticaly evaluate influence of the exponential decay of magnetic field on the fate of isolated neutron stars. The fact of ROSAT observations of several X-ray sources, which can be accreting old isolated neutron stars gives us an opportunity to put some limits on the parameters of the exponential decay. We argue, that, if most part of neutron stars have approximately the same decay and initial parameters, then the combinations of the bottom magnetic momentum, $\mu_b$, in the range $\sim 10^{28}-10^{29.5} {\rm G} {\rm cm}^3$ and characteristic time scale, $t_d$, in the range $\sim 10^7-10^8 {\rm yrs}$ for standard initial magnetic momentum, $\mu_0=10^{30} {\rm G} {\rm cm}^3$, can be excluded, because for that sets of parameters neutron stars never come to the stage when accretion of the interstellar medium on their surfaces is possible even for low velocity of neutron stars and relatively high density of the interstellar medium. The region of excluded parameters increases with $\mu_0$ decreasing.Comment: 5 pages, 4 PostScript figures (uses A&A style