The number of non-solutions to an equation in a group and non-topologizable torsion-free groups

It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of non-solutions to this equation. A countable torsion-free non-topologizable group is constructed.Comment: 5 pages; minor changes in the introduction and reference

Reflection principle characterizing groups in which unconditionally closed sets are algebraic

We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63 years old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G.Comment: 14 page

Optimal control over geomorphological systems

The method presented can be used for the planning of shore protection measures without seawalls or other cliff-protecting structures; it is particularly appropriate in cases where the system is eventually to control itself by adjustment to a dynamic equilibrium regim

Skeletal Anomalies of Juvenile Smooth Newts - Lissotriton vulgaris

An article contains first data on internal skeletal anomalies of smooth newt juveniles. The material was collected on the territory of urban agglomeration of Yekaterinburg. We found 16 variants of anomalies. An increase of general frequency of deviations along urbanization gradient were mentioned. The differences and specificity ofthe populations spectra are related with the profiling of the juveniles definitive morphology by environmental specifics

Deep CDP seismic studies in the Volga-Ural petroleum province

Deep CDP seismic studies were initiated in the Volga-Ural petroleum province in 1993 on two lines crossing the Romashkino giant field. The deep seismic studies aimed to identify differences between the structure of the Earth's crust in the largest oil field of the Volga-Ural province and that outside of it. The data acquired in 1994-95 unequivocally indicated that these differences did exist. The existing concepts of the major role played by the Earth's deep crust in the distribution of hydrocarbon deposits could be further developed and implemented through a geotraverse survey across the Volga - Ural petroliferous province. The detection of relationships between the petroleum potential of sedimentary cover and the structure of the crust and mantle is of fundamental significance (deep origin of hydrocarbons). Specificance;c features revealed in this region can be implemented for oil prospecting in the Volga-Ural and other regions, as well as for the assessment of oil prospects in poorly studied regions

Three natural mechanical systems on Stiefel varieties

We consider integrable generalizations of the spherical pendulum system to the Stiefel variety $V(n,r)=SO(n)/SO(n-r)$ for a certain metric. For the case of V(n,2) an alternative integrable model of the pendulum is presented. We also describe a system on the Stiefel variety with a four-degree potential. The latter has invariant relations on $T^*V(n,r)$ which provide the complete integrability of the flow reduced on the oriented Grassmannian variety $G^+(n,r)=SO(n)/SO(r)\times SO(n-r)$.Comment: 14 page

Mathematical simulation of stable and equilibrium river bed profiles and slopes.

The first approach deals with a solution to the equation of solid matter (sediment) continuity in the equilibrium case and the second approach deals with the application of variational principles. The approaches considered and results obtained can be used in hydroengineering construction for designing stable and equilibrium longitudinal profiles of water streams and slopes.-from Author