File bases in groups

File bases — special systems of group generators generalizing the usual bases of Abelian groups — are considered. Examples of such bases in different group classes are listed. It is shown that if some initial groups have file bases, then direct and free products of these groups as well as their extensions also have such bases. With some additional restrictions imposed, the existence of a file basis in factor groups is proved. A list of problems is offered. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 247-261, May-June, 1995.     

Whispering-gallery modes in a barrel-shaped dielectric resonator

We experimentally study a new type of resonator, namely, a barrel-shaped dielectric resonator with whispering-gallery modes which is formed by the cutting dielectric hemisphere. Compared with a hemispherical dielectric resonator, the E type oscillation spectrum of such a resonator is rarefied with respect to the azimuthal index. Comparative analysis of the electromagnetic characteristics of the barrel-shaped dielectric resonator and half-disk dielectric resonators shows that the radiation losses of the mode energy from the spherical surface are smaller than those from the cylindrical surface. This fact stimulates the high values of the internal Q-factor of modes of the barrel-shaped dielectric resonator. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 12, pp. 1041–1048, December 2005.     

Group Quasivarieties with Infinitely Many Maximal Subquasivarieties

We construct an example of a 2-generated group G and describe a set of proper maximal subquasivarieties in the quasivariety qG. This set proves to be infinite, which gives an affirmative answer to Question 14.25 posed by A. I. Budkin in the Kourovka Notebook. Moreover, it is shown that every proper subquasivariety in qG is contained in a proper maximal one.     

Completion of Linearly Ordered Metabelian Groups

We prove a theorem saying that in finitely generated linearly ordered metabelian groups there exists a finite system of normal convex subgroups satisfying orderability conditions for groups, and an embedding theorem for linearly ordered metabelian groups whose initial linear orders extend to -divisible linearly ordered metabelian ones. As a consequence, it is stated that orderable metabelian groups are embedded, with extension of all their linear orders, in -divisible orderable metabelian groups.     

On Ordering the Groups with Nilpotent Commutant

We prove that every group with nilpotent commutant, having an abelian normal subgroup such that the factor by this subgroup is nilpotent, is preorderable if and only if the group is -torsion-free. An example is exhibited of a nonorderable -torsion-free group with two-step nilpotent radical. This example demonstrates that for the variety of groups with nilpotent commutant the absence of -torsion in a group is not a sufficient condition for orderability.     

Locally nilpotent groups with the minimal condition on centralizers

It is proved that locally nilpotent groups with the minimal condition on centralizers are hypercentral, and that the Fitting subgroup of a group with the minimal condition on centralizers of normal subgroups is nilpotent. Supported by RFFR grant No. 96-01-00358. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 270–278, May–June, 1998.     

Right orders and amalgamation for lattice-ordered groups

Let H _i be a sublattice subgroup of a lattice-ordered group G _i (i = 1, 2). Suppose that H ₁ and H ₂ are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G ₁ and G ₂ can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H ₁ and H ₁ ^φ . In this article we prove that the group free product of G ₁ and G ₂ amalgamating H ₁ and H ₁ ^φ is right orderable and so embeddable (as a group) in a lattice-orderable group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable [BLUDOV, V. V.—GLASS, A. M. W.: Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. (3) 99 (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal (?-ideals) and solve of Problem 1.42 from [KOPYTOV, V. M.—MEDVEDEV, N. YA.: Ordered groups. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].     

Long-living BLOCH oscillations of matter waves in periodic potentials

The dynamics of matter waves in linear and nonlinear optical lattices subject to a spatially uniform linear force is studied both analytically and numerically. It is shown that by properly designing the spatial dependence of the scattering length it is possible to induce long-living Bloch oscillations of gap-soliton matter waves in optical lattices. This occurs when the effective nonlinearity and the effective mass of the soliton have opposite signs for all values of the crystal momentum in the Brillouin zone. The results apply to all systems modeled by the periodic nonlinear Schr?dinger equation, including propagation of light in photonic and photorefractive crystals with tilted band structures.     

Thresholdless surface solitons

We report on the existence of nonlinear surface waves that, on the one hand, do not require the threshold energy flow for their excitation, and, on the other hand, extend into media at both sides of the interface at low powers, i.e., cannot be reduced to the conventional Tamm states. Such waves can be excited if the refractive index in at least one of the materials forming the interface is periodically modulated, with properly selected modulation depth and frequency. Thresholdless surface solitons can be stable in the entire existence domain.