The inverse problem for the Gross - Pitaevskii equation

Two different methods are proposed for the generation of wide classes of exact solutions to the stationary Gross - Pitaevskii equation (GPE). The first method, suggested by the work by Kondrat'ev and Miller (1966), applies to one-dimensional (1D) GPE. It is based on the similarity between the GPE and the integrable Gardner equation, all solutions of the latter equation (both stationary and nonstationary ones) generating exact solutions to the GPE, with the potential function proportional to the corresponding solutions. The second method is based on the "inverse problem" for the GPE, i.e. construction of a potential function which provides a desirable solution to the equation. Systematic results are presented for 1D and 2D cases. Both methods are illustrated by a variety of localized solutions, including solitary vortices, for both attractive and repulsive nonlinearity in the GPE. The stability of the 1D solutions is tested by direct simulations of the time-dependent GPE

Recognizability by Prime Graph of the Group 2 E 6(2)

It is proved that the simple group 2E6(2) is recognized by its prime graph. © 2021, Springer Science+Business Media, LLC, part of Springer Nature

Thermal and Dynamical Equilibrium in Two-Component Star Clusters

We present the results of Monte Carlo simulations for the dynamical evolution of star clusters containing two stellar populations with individual masses m1 and m2 > m1, and total masses M1 and M2 < M1. We use both King and Plummer model initial conditions and we perform simulations for a wide range of individual and total mass ratios, m2/m1 and M2/M1. We ignore the effects of binaries, stellar evolution, and the galactic tidal field. The simulations use N = 10^5 stars and follow the evolution of the clusters until core collapse. We find that the departure from energy equipartition in the core follows approximately the theoretical predictions of Spitzer (1969) and Lightman & Fall (1978), and we suggest a more exact condition that is based on our results. We find good agreement with previous results obtained by other methods regarding several important features of the evolution, including the pre-collapse distribution of heavier stars, the time scale on which equipartition is approached, and the extent to which core collapse is accelerated by a small subpopulation of heavier stars. We briefly discuss the possible implications of our results for the dynamical evolution of primordial black holes and neutron stars in globular clusters.Comment: 31 pages, including 13 figures, to appear in Ap

ASD moduli spaces over four-manifolds with tree-like ends

In this paper we construct Riemannian metrics and weight functions over Casson handles. We show that the corresponding Atiyah-Hitchin-Singer complexes are Fredholm for some class of Casson handles of bounded type. Using these, the Yang-Mills moduli spaces are constructed as finite dimensional smooth manifolds over Casson handles in the class.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper21.abs.htm

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and Hg are conjugate in 〈H,Hg〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSLn(q), PSUn(q), E6(q), 2E6(q), where in all cases q is odd and n is not a power of 2, and P Sp2n(q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp2n(q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups. The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp2n(q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2m or 2m(22k+1), this group has a nonpronormal subgroup of odd index. If n = 2m, then we show that all subgroups of P Sp2n(q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp2n(q) is still open when n = 2m(22k + 1) and q ≡ ±3 (mod 8). © 2017, Pleiades Publishing, Ltd

RECOGNITION OF THE GROUP E6(2) BY GRUENBERG–KEGEL GRAPH

The Gruenberg–Kegel graph (or the prime graph) of a finite group G is a simple graph Γ(G) whose vertices are the prime divisors of the order of G, and two distinct vertices p and q are adjacent in Γ(G) if and only if G contains an element of order pq. A finite group is called recognizable by Gruenberg–Kegel graph if it is uniquely determined up to isomorphism in the class of finite groups by its Gruenberg–Kegel graph. In this paper, we prove that the finite simple exceptional group of Lie type E6(2) is recognizable by its Gruenberg–Kegel graph. © 2021 The authors.National Natural Science Foundation of China, NSFC, (12171126)Ministry of Education and Science of the Russian Federation, MinobrnaukaWu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, (075-02-2021-1387)The work is supported by the National Natural Science Foundation of China (project No. 12171126), by Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences, and by the Regional Scientific and Educational Mathematical Center “Ural Mathematical Center” under the agreement No. 075-02-2021-1387 with the Ministry of Science and Higher Education of the Russian Federation

On the pronormality of subgroups of odd index in finite simple groups

We prove the pronormality of subgroups of finite index for many classes of simple groups. © 2015, Pleiades Publishing, Ltd

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem $-\Delta u = f$, u |\_{\pa \PP} = g on a polyhedral domain \PP \subset \RR^3 using the \BK\ spaces \Kond{m}{a}(\PP). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges \cite{Babu70, Kondratiev67}. In particular, we show that there is no loss of \Kond{m}{a}--regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in \Kond{m}{a}(\PP)

Giant Pulses -- the Main Component of the Radio Emission of the Crab Pulsar

The paper presents an analysis of dual-polarization observations of the Crab pulsar obtained on the 64-m Kalyazin radio telescope at 600 MHz with a time resolution of 250 ns. A lower limit for the intensities of giant pulses is estimated by assuming that the pulsar radio emission in the main pulse and interpulse consists entirely of giant radio pulses; this yields estimates of 100 Jy and 35 Jy for the peak flux densities of giant pulses arising in the main pulse and interpulse, respectively. This assumes that the normal radio emission of the pulse occurs in the precursor pulse. In this case, the longitudes of the giant radio pulses relative to the profile of the normal radio emission turn out to be the same for the Crab pulsar and the millisecond pulsar B1937+21, namely, the giant pulses arise at the trailing edge of the profile of the normal radio emission. Analysis of the distribution of the degree of circular polarization for the giant pulses suggests that they can consist of a random mixture of nanopulses with 100% circular polarization of either sign, with, on average, hundreds of such nanopulses within a single giant pulse.Comment: 13 pages, 6 figures (originally published in Russian in Astronomicheskii Zhurnal, 2006, vol. 83, No. 1, pp. 62-69) translated by Denise Gabuzd