We analyze a minimum number of vertices of a complete graph that can be decomposed into one factor of diameter 2 and k factors of diameter at most 3. We find exact values for k ≤ 4 and the asymptotic value of the ratio of this number and k when k tends to infinity. We also find the asymptotic value of the ratio of the number of vertices of the smallest complete graph that can be decomposed into p factors of diameter 2 and k factors of diameter 3 and number k when p is fixed.

We present results on numerical regulator design for sampled-data nonlinear plants via their approximate discrete-time plant models. The regulator design is based on an approximate discrete-time plant model and is carried out either via an infinite horizon optimization problem or via a finite horizon with terminal cost optimization problem. In both cases, we discuss situations when the sampling period T and the integration period h used in obtaining the approximate discrete-time plant model are the same or they are independent of each other. We show that, using this approach, practical and/or semiglobal stability of the exact discrete-time model is achieved under appropriate conditions.

Air coolers consume much more energy compared to other heat exchangers due to the large fan power required. This is an additional reason to establish reliable methods for the rational design and thermohydraulic analysis of these devices. The optimal values of the outlet temperature and air flow rate are of particular importance. The paper presents a methodology for the thermohydraulic calculation of air cooler performances, which is incorporated in the "Air Cooler" software module. The module covers two options: cooling and/or condensation of process fluids by ambient air. The calculated results can be given in various ways ie. in the tabular and graphical form.

The determination of the optimal process conditions for the operation of air coolers demands a detailed analysis of their thermohydraulic behaviour on the one hand, and the estimation of the operating costs, on the other. One of the main parameters of the thermohydraulic behaviour of this type of equipment, is the outlet air temperature. The influence of the outlet air temperature on the performance of air coolers (heat transfer coefficient overall heat transfer coefficient, required surface area for heat transfer air-side pressure drop, fan power consumption and sound pressure level) was investigated in this study. All the computations, using AirCooler software [1], were applied to cooling of the process fluid and the condensation of a multicomponent vapour mixture on two industrial devices of known geometries.