On the cyclic subgroup separability of the free product of two groups with commuting subgroups

Let G be the free product of groups A and B with commuting subgroups H \leqslant A and K \leqslant B, and let C be the class of all finite groups or the class of all finite p-groups. We derive the description of all C-separable cyclic subgroups of G provided this group is residually a C-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p'-isolated in the free factors, then all p'-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p'-isolated and cyclic

A characterization of root classes of groups

A class of groups C is root in a sense of K. W. Gruenberg if it is closed under taking subgroups and satisfies the Gruenberg condition: for any group X and for any subnormal sequence Z \leqslant Y \leqslant X with factors in C, there exists a normal subgroup T of X such that T \leqslant Z and X/T \in C. We prove that a class of groups is root if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we prove also that, if C is a nontrivial root class of groups closed under taking quotient groups and G = is the generalized free product of two nilpotent C-groups A and B possessing \varphi-compartible central series, then G is residually a solvable C-group.Comment: This is an Author's Original Manuscript of an article submitted for consideration in the "Communications in Algebra" [copyright Taylor & Francis]; "Communications in Algebra" is available online at http://www.tandfonline.co

On the cyclic subgroup separability of free products of two groups with amalgamated subgroup

Let $G$ be a free product of two groups with amalgamated subgroup, $\pi$ be either the set of all prime numbers or the one-element set \{$p$\} for some prime number $p$. Denote by $\Sigma$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $\pi$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $\pi$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $\pi^{\prime}$-isolated, don't belong to $\Sigma$. Some sufficient conditions are obtained for $\Sigma$ to coincide with the family of all other $\pi^{\prime}$-isolated cyclic subgroups of group $G$. It is proved, in particular, that the residual $p$-finiteness of a free product with cyclic amalgamation implies the $p$-separability of all $p^{\prime}$-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$-finite nilpotent groups.Comment: 10 pages; for other papers of this author, see http://icu.ivanovo.ac.ru/tg-semina

q-Bosonization of the quantum group GLq(2) GL_q(2) based on the Gauss decomposition

The new method of q-bosonization for quantum groups based on the Gauss decomposition of a transfer matrix of generators is suggested. The simplest example of the quantum group $GL_q(2)$ is considered in some details.Comment: 10 pages, LaTe

Five-loop renormalization-group expansions for two-dimensional Euclidean \lambda \phi^4 theory

The renormalization-group functions of the two-dimensional n-vector \lambda \phi^4 model are calculated in the five-loop approximation. Perturbative series for the \beta-function and critical exponents are resummed by the Pade-Borel-Leroy techniques. An account for the five-loop term shifts the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the lattice calculations. This is argued to reflect the influence of the non-analytical contribution to the \beta-function. The evaluation of the critical exponents for n = 1, n = 0 and n = -1 in the five-loop approximation and comparison of the results with known exact values confirm the conclusion that non-analytical contributions are visible in two dimensions. For the 2D Ising model, the estimate \omega = 1.31 for the correction-to-scaling exponent is found.Comment: LaTeX, 7 pages, 3 table

On differential operators for bivariate Chebyshev polynomials

We construct the differential operators for which bivariate Chebyshev polynomials of the first kind, associated with simple Lie algebras $C_2$ and $G_2$, are eigenfunctions

Critical thermodynamics of the two-dimensional systems in five-loop renormalization-group approximation

The RG functions of the 2D $n$-vector $\phi^4$ model are calculated in the five-loop approximation. Perturbative series for the $\beta$ function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $\beta$ function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions.Comment: 8 pages, 3 tables, as publishe

The cyclic subgroup separability of certain generalized free products of two groups

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups

Renormalized sextic coupling constant for the two-dimensional Ising model from field theory

The field-theoretical renormalization group approach is used to estimate the universal critical value g_6^* of renormalized sextic coupling constant for the two-dimensional Ising model. Four-loop perturbative expansion for g_6 is calculated and resummed by means of the Pade-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure the estimates g_6^* = 1.10, g_6^*/{g_4^*}^2 = 2.94 are obtained which agree quite well with those deduced recently by S.-Y. Zinn, S.-N. Lai, and M. E. Fisher (Phys. Rev. E 54 (1996) 1176) from the high-temperature expansions.Comment: 8 pages, LaTeX, no figures, published versio

Centers of near-infrared luminescence in bismuth-doped TlCl and CsI crystals

A comparative first-principles study of possible bismuth-related centers in TlCl and CsI crystals is performed and the results of computer modeling are compared with the experimental data. The calculated spectral properties of the bismuth centers suggest that the IR luminescence observed in TlCl:Bi is most likely caused by Bi--Vac(Cl) centers (Bi^+ ion in thallium site and a negatively charged chlorine vacancy in the nearest anion site). On the contrary, Bi^+ substitutional ions and Bi_2^+ dimers are most likely responsible for the IR luminescence observed in CsI:Bi.Comment: 8 pages, 4 figure