The paper proposes an artificial neural network (ANN) being a global approximator for a special class of functions, which are known as generalized homogeneous. The homogeneity means a symmetry of a function with respect to a group of transformations having topological characterization of a dilation. In this paper, a class of the so-called linear dilations is considered. A homogeneous universal approximation theorem is proven. Procedures for an upgrade of an existing ANN to a homogeneous one are developed. Theoretical results are supported by examples from the various domains (computer science, systems theory and automatic control).

We study the cutwidth measure on graphs and ways to bound the cutwidth of a graph by partitioning its vertices. We consider bounds expressed as a function of two quantities: on the one hand, the maximal cutwidth x of the subgraphs induced by the classes of the partition, and on the other hand, the cutwidth y of the quotient multigraph obtained by merging each class to a single vertex. We consider in particular decomposition of directed graphs into strongly connected components (SCCs): in this case, x is the maximal cutwidth of an SCC, and y is the cutwidth of the directed acyclic condensation multigraph. We show that the cutwidth of a graph is always in O(x + y), specifically it can be upper bounded by 1.5x + y. We also show a lower bound justifying that the constant 1.5 cannot be improved in general