Estimate of the Number of Edges in Subgraphs of a Johnson Graph

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained.

Two-Level Regression Method Using Ensembles of Trees with Optimal Divergence

The article discusses a new two-level regression analysis method in which a corrective procedure is applied to optimal ensembles of regression trees. Optimization is carried out based on the simultaneous achievement of the divergence of the algorithms in the forecast space and a good approximation of the data by individual algorithms of the ensemble. Simple averaging, random regression forest, and gradient boosting are used as corrective procedures. Experiments are presented comparing the proposed method with the standard decision forest and the standard gradient boosting method for decision trees.

Symplectic Geometry of the Koopman Operator

We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.

Covering Planar Sets

Methods for improving upper and lower bounds for various coverings of planar sets are proposed. New bounds for various numbers of partition constituents are presented, and suggestions for the generalization of the presented methods are offered.