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This research details the approach to justify optimum thermal conditions for a vehicle transmission unit. This approach is based on the integrated criterion which comprises aggregated energy expenditures to warm up a transmission reduction gear and to overcome forces of mechanical and hydraulic friction inside its components. Thermal working conditions of a unit shall be considered optimum when energy expenditures are the lowest. Justification and development of technologies to provide optimum thermal conditions of a unit shall be based on comparison of optimum temperature values and temperature values of transmission oil stabilization. If optimum temperature exceeds stabilization temperature, then additional heat sources shall be used if it is lower then heat shall be transferred to other units. Analysis of optimum thermal conditions of the KAMAZ transmission shows that the transmission unit has an excessive thermal potential. Optimum temperature exceeds stabilization temperature by 5 K, so, excessive thermal energy must be transferred.

Let SB n be the singular braid group generated by braid generators σ i and singular braid generators τ i , 1 ≤ i ≤ n − 1. Let ST n denote the group that is the kernel of the homomorphism that maps, for each i, σ i to the cyclic permutation (i, i + 1) and τ i to 1. In this paper we investigate the group ST 3 . We prove that ST 3 is isomorphic to the pure singular braid group SP 3 on 3 strands.

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