The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces

The weak Bieberbach theorem states that each crystallographic group on a Euclidean space uniquely determines its translation lattice as an abstract group. Garipov proved in 2003 that the same holds for crystallographic groups on Minkowski spaces and asked whether a similar claim holds in the pseudo-Euclidean spaces ℝ ^p,q . We prove that the weak Bieberbach theorem holds for crystallographic groups on pseudo-Euclidean spaces ℝ ^p,q with min{p, q} ≤ 2. For min{p, q} ≥ 3 we construct examples of crystallographic groups with two distinct lattices exchanged by a suitable automorphism of the group. For crystallographic groups with two distinct isomorphic pseudo-Euclidean lattices we also prove that the coranks of their intersection in these lattices can take arbitrary values greater than 2 with the exception of 4.     

Synchrotron Radiation Facility “Siberian Circular Photon Source” (SRF SKIF)

Crystallography Reports - The main technical solutions chosen to implement the project of fourth-generation synchrotron radiation (SR) source Synchrotron Radiation Facility “Siberian Circular...     

The portion of matrices with real spectrum in the real orthogonal Lie algebra

The portion of matrices with real spectrum in a matrix Lie algebra is the ratio of the volume of the set of matrices with real spectrum in a ball centered at the zero of the algebra to the volume of the whole ball. We calculate the portion for the real orthogonal Lie algebra.     

Pulsed inductive discharge CO2 laser

A pulsed inductive discharge CO₂ laser with a wavelength of 10.6 μm has been created for the first time. The excitation system of a cylindrical pulsed inductive discharge (pulsed inductively coupled plasma) in the gas mixture of CO₂:N₂:He was developed. The temporal and energy parameters of the laser radiation were investigated. The maximum inductive discharge CO₂ laser radiation energy of 104 mJ was achieved. An average power of 3.2 W was obtained at laser generation energy of 65 mJ and pulse repetition rate of 50 Hz. In the cross-section, the laser radiation had the ring shape with an external diameter of 34 mm and thickness of 4-5 mm. The measured divergence of laser radiation was 12 mrad.     

Statistical properties of partially coherent cw fiber lasers

We perform a detailed quantitative numerical analysis of a partially coherent quasi-cw fiber laser on the example of a high-Q normal dispersion cavity Raman fiber laser. The key role of precise spectral performances of fiber Bragg gratings forming the laser cavity is clarified. It is shown that cross-phase modulation between the pump and Stokes waves does not affect the generation. Amplitudes of different longitudinal modes strongly fluctuate, obeying the Gaussian distribution. As the intensity statistics is noticeably nonexponential, longitudinal modes should be correlated.     

703-to 731-nm FI laser excited by a transverse inductive discharge

It is proposed to use a pulsed transverse inductive discharge for exciting gas lasers operating on electron transitions in atoms and molecules. An electron transition laser on fluorine (FI) atoms pumped by a transverse inductive discharge is developed. Lasing at three wavelengths (703.75, 712.79, and 731.1 nm) is obtained by exciting He-F₂ (NF₃) gas mixtures in a pressure range from 20 to 350 Torr. The results of experimental investigation of the spectral, temporal, and energy characteristics of the inductive FI laser are presented.     

A continuous version of the Hausdorff–Banach–Tarski paradox

We come up with a simple proof for a continuous version of the Hausdorff–Banach–Tarski paradox, which does not make use of Robinson’s method of compatible congruences and fits in the case of finite and countable paradoxical decompositions. It is proved that there exists a free subgroup whose rank is of the power of the continuum in a rotation group of a three-dimensional Euclidean space. We also argue that unbounded subsets of Euclidean space containing inner points are denumerably equipollent.