A new proof of Vassiliev's conjecture

We give a new proof of Vassiliev's planarity criterion for framed four-valent graphs (and more generally, *-graphs), which is based on Pontryagin-Kuratowski theorem.Comment: a planarity criterion for noneven *-graphs is adde

On the microeconomic problems studied by portfolio theory

In the paper we consider economically motivated problems, which are treated with the help of methods of portfolio theory that goes back to the papers by H. Markowitz [1] and J. Tobin [2]. We show that the portfolio theory initially developed for risky securities (stocks) could be applied to other objects. In the present paper we consider several situations where such an application is reasonable and seems to be fruitful. Namely, we consider the problems of constructing the efficient portfolio of banking services and the portfolio of counteragents of a firm. © 2012 American Institute of Physics