It is important to work out precise and easy-to-use mathematical models to predict the indoor temperature in buildings for human residence. Such models can support model-based/predictive controls to efficiently maintain the temperature at a comfortable level.

The proper allocation/distribution of limited resources is a traditional problem with various applications. The mathematical formulation of such problems usually includes constraints describing the set of feasible solutions (feasible set), from which the (nearly) optimal or equilibrium solution should be selected. Often the feasible set is more difficult to determine than to find the optimal or equilibrium solution. Alternatively, the already known feasible set often makes it easier to select the optimal or equilibrium solution. In some other cases, any feasible solutions are the same satisfactory, additional optimization is needless. Accordingly, the main or only task in many cases is to determine the feasible set itself. In the paper, a new theorem is proved for the explicit expression of properly assigned (dependent) variables by means of the other (independent) variables in a system of inequality and quadratic equality constraints. The sum of the (nonnegative) variables can be either prefixed or not. The constraints may describe the feasible set in various resource allocation tasks (possibly in optimization or game-theoretical contexts) or in other problems. Two new lemmas are proved for supporting the proof of the above mentioned theorem, nevertheless, they can also be considered independent results, which may help future mathematical derivations. Supported by a further new lemma, a practical algorithm is derived for assigning in a feasible way the independent variables, to which (possibly limited) arbitrary nonnegative values can be prescribed. Various practical examples are provided to facilitate utilizing the results.

Bimatrix games have had theoretical importance and important applications since the very beginning of game theory to date. Pareto optimal strategy vectors represent a reasonable and frequently applied solution concept (basically in the cooperative approach). Particularly, it often arises whether a non-cooperative solution can be improved in cooperative sense, i.e. it is Pareto optimal or not. However, in concrete cases it may be hard to determine the Pareto optimal strategy vectors or at least check the Pareto optimality of a given strategy vector. In the present paper, the Pareto optimality of the strategy pairs in general n × m ( $$n = 2,3, \ldots$$n = 2 , 3 , … ; $$m = 2,3, \ldots$$m = 2 , 3 , … ) bimatrix games is studied. First of all an elementary proof is provided for a theorem, which makes the proposed Pareto optimality checking algorithm simpler. Then a nonlinear transform is proposed, which makes the algorithm even more convenient in the important case of 2 × 2 bimatrix games (and for certain other cases). Two numerical examples present the practical applicability of the checking algorithm. The problem of 2 × 2 bimatrix games can be solved even by hand. For larger games, numerous computational tools are available in the practice to apply the checking algorithm in exact or at least approximate way.