Balanced factorisations

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.Comment: 7 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.ht

Relaxation of dark matter halos: how to match observational data?

We show that moderate energy relaxation in the formation of dark matter halos invariably leads to profiles that match those observed in the central regions of galaxies. The density profile of the central region is universal and insensitive to either the seed perturbation shape or the details of the relaxation process. The profile has a central core; the multiplication of the central density by the core radius is almost independent of the halo mass, in accordance with observations. In the core area the density distribution behaves as an Einasto profile with low index ($n\sim 0.5$); it has an extensive region with $\rho\propto r^{-2}$ at larger distances. This is exactly the shape that observations suggest for the central region of galaxies. On the other hand, this shape does not fit the galaxy cluster profiles. A possible explanation of this fact is that the relaxation is violent in the case of galaxy clusters; however, it is not violent enough when galaxies or smaller dark matter structures are considered. We discuss the reasons for this.Comment: 9 pages, 4 figures, accepted to Astronomy & Astrophysic

Test of multiscaling in DLA model using an off-lattice killing-free algorithm

We test the multiscaling issue of DLA clusters using a modified algorithm. This algorithm eliminates killing the particles at the death circle. Instead, we return them to the birth circle at a random relative angle taken from the evaluated distribution. In addition, we use a two-level hierarchical memory model that allows using large steps in conjunction with an off-lattice realization of the model. Our algorithm still seems to stay in the framework of the original DLA model. We present an accurate estimate of the fractal dimensions based on the data for a hundred clusters with 50 million particles each. We find that multiscaling cannot be ruled out. We also find that the fractal dimension is a weak self-averaging quantity. In addition, the fractal dimension, if calculated using the harmonic measure, is a nonmonotonic function of the cluster radius. We argue that the controversies in the data interpretation can be due to the weak self-averaging and the influence of intrinsic noise.Comment: 8 pages, 9 figure

Why does Einasto profile index n∼6n\sim 6 occur so frequently?

We consider the behavior of spherically symmetric Einasto halos composed of gravitating particles in the Fokker-Planck approximation. This approach allows us to consider the undesirable influence of close encounters in the N-body simulations more adequately than the generally accepted criteria. The Einasto profile with index $n \approx 6$ is a stationary solution of the Fokker-Planck equation in the halo center. There are some reasons to believe that the solution is an attractor. Then the Fokker-Planck diffusion tends to transform a density profile to the equilibrium one with the Einasto index $n \approx 6$. We suggest this effect as a possible reason why the Einasto index $n \approx 6$ occurs so frequently in the interpretation of N-body simulation results. The results obtained cast doubt on generally accepted criteria of N-body simulation convergence.Comment: 7 pages, 2 figures, Accepted to JCA