For almost two decades, monotonic, or "delete free," relaxation has been one of the key auxiliary tools in the practice of domain-independent deterministic planning. In the particular contexts of both satisficing and optimal planning, it underlies most state-of-theart heuristic functions. While satisficing planning for monotonic tasks is polynomial-time, optimal planning for monotonic tasks is NP-equivalent. Here we establish both negative and positive results on the complexity of some wide fragments of optimal monotonic planning, with the fragments being defined around the causal graph topology. Our results shed some light on the link between the complexity of general optimal planning and the complexity of optimal planning for the respective monotonic relaxations.
Complexity of the agricultural business tasks, high dynamism and non-linear nature of the contemporary socio-economic processes which differs functioning of any industry are placing new requirements for predictive studies. The purpose of this study is to develop a set of methodological provisions for the construction of predictive scenarios of the agricultural business by identifying current trends, the impact factors of the environment and the interpretation of results forecasting and analytical calculations. This article considers the influence of climatic factors on the economic impact of the frumentaceous and the grape branches of agriculture. The system of economic and mathematical prognostics models of the main industrial indicators was developed. The methodology for scenario forecasting of indicators of frumentaceous production and vine growing was proposed based on the use of the influence of solar activity on agrobiological processes.
Mathematical induction is one of the main methods of the proof that students study and use it in their studies in higher education. This important method, in addition to mathematics, is widely used in computer science and a number of other related disciplines. However, even if the principles of proof using the method of induction are taught, understood and mastered by students well, many students have great difficulty with algebraic transformations necessary for proving, for example, finding the inductive step. The use of the Maple computer algebra system allows students to overcome this obstacle and, more attention and effort to pay to understanding the concept of proofs using the method of mathematical induction. In addition, students can prove more complex algebraic statements, and their activities can be of a pronounced research nature.
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