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We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the ‘giant component’ and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.

Article Info Received : 07.08.2015 Accepted : 09.06.2016 The goal of this study was to determine the conductive and steam-diffusive heat transfer constituents in the soil. Based on the solution of differential equation system of heat and mass transfer, the method to determine the conductive and steam-diffusive heat transfer constituents in wet soils was developed. To measure the thermophysical properties in laboratory setting, a pulse method of a two-dimensional heat source was used. The method takes into account the patterns of temperature field equalization in an unbounded medium after the heat source termination. A feature of this process is the occurrence of peak temperature at the investigated point of the medium at a given instant. In this experiment, the temperature was controlled not only at the investigated point of the medium, but also at the soil-heater interface. The proposed method was used to study the thermophysical indices of the chernozems of the Altai Region's Priobye area (the Ob River area). The chernozem under study is of light-loamy particlesize composition; the illuvial horizon В is enriched by a sand fraction. It has been found that the soil conductive thermal diffusivity is reduced with increasing moisture content. The steam-diffusive thermal diffusivity has the extremum at the moisture close to the discontinuous capillary moisture. In humus horizons it plays a smaller role than in the mineral horizons. The thermal diffusivity determined by the steam molecule motion in the pore space of the soil exceeds the conductive thermal diffusivity two or three times. At the same time thermal steam diffusivity is more than ten time lower than the conductive constituent. Eventually, the stem molecules though dramatically accelerating the processes of heat transfer in the soil profile conduct a small amount of heat and make a weak contribution to soil thermal accumulation.

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