How to Compute Spectra with Error Control.

Computing the spectra of operators is a fundamental problem in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve. Moreover, they are easy to implement and parallelize, offer fundamental speed-ups, and allow problems that before, regardless of computing power, were out of reach. Results are demonstrated on difficult problems such as the spectra of quasicrystals and non-Hermitian phase transitions in optics.This work was supported by Engineering and Physical Sciences Research Council Grants No. EP/L016516/1, No. EP/R008272/1, No. EP/N014588/1, and No. EP/ L003457/1, as well as a Royal Society University Research Fellowship

On the supersymmetric partition function in QCD-inspired random matrix models

We show that the expression for the supersymmetric partition function of the chiral unitary (Laguerre) ensemble conjectured recently by Splittorff and Verbaarschot [Phys. Rev. Lett. 90, 041601 (2003)] follows from the general expression derived recently by Fyodorov and Strahov [J. Phys. A: Math. Gen. 36, 3203 (2003)]

Computer graphics in course engineering graphs for professions 250201, 250203, 0302

Рассмотрены проблемы преподавания курса инженерной графики и возможности компьютерных технологий.Considered problems of the teaching the course engineering graphs and possibility computer technology

Signatures of Interstellar-Intracluster Medium Interactions: Spiral Galaxy Rotation Curves in Abell 2029

We investigate the rich cluster Abell 2029 (z~0.08) using optical imaging and long-slit spectral observations of 52 disk galaxies distributed throughout the cluster field. No strong emission-line galaxies are present within ~400 kpc of the cluster center, a region largely dominated by the similarly-shaped X-ray and low surface brightness optical envelopes centered on the giant cD galaxy. However, two-thirds of the galaxies observed outside the cluster core exhibit line emission. H-alpha rotation curves of 14 cluster members are used in conjunction with a deep I band image to study the environmental dependence of the Tully-Fisher relation. The Tully-Fisher zero-point of Abell 2029 matches that of clusters at lower redshifts, although we do observe a relatively larger scatter about the Tully-Fisher relation. We do not observe any systematic variation in the data with projected distance to the cluster center: we see no environmental dependence of Tully-Fisher residuals, R-I color, H-alpha equivalent width, and the shape and extent of the rotation curves.Comment: 22 pages, 6 figures, 3 tables; to appear in the August 2000 Astronomical Journa