### Working paper

## Две модели принятия решений участником торгов на фондовой бирже по формированию и изменению своего инвестиционного портфеля

The paper discusses a new approach to developing tools for quantitatively analyzing the financial behavior of small and medium price-taking traders each possessing abilities to predict share price values for a set of financial securities traded in a stock exchange. Tools for forming and managing a trader’s portfolio of securities from this set are proposed. Particularly, it is shown that when the trader can treat share price values from the portfolio as random variables with known (to her) distributions, an optimal portfolio composition is found by solving a linear programming problem. Otherwise, this optimal composition is found as the trader’s equilibrium strategy in an antagonistic two-person game with the stock exchange being the other player. In this game on polyhedra of disjoint player strategies, described by systems of linear equations and inequalities of a balance kind, calculating saddle points is reduced to solving linear programming problems forming a dual pair.

The issue of using the MathCAD software package in a university educational course for learning to solve optimization problems is considered. The advantage of working with this program is shown and its main features are discussed in the appendix to this course.

Contents of the book is divided into 2 parts of deterministic and stochastic models of Operations Research.

The first part of "Deterministic models of Operations Research" - is the base section, in which the emphasis is on linear programming.

The second part - "Stochastic models of Operations Research" includes a model of reliability and queuing models. This is original material.

The textbook can be useful to students of undergraduate and graduate programs in areas of training in "Applied Mathematics", "Applied Mathematics and Computer Science", "Information systems and technologies", as well as graduate students and science teachers who are interested in the problems of optimization in stochastic models

Ionization processes for a two dimensional quantum dot subjected to combined electrostatic and alternating electric fields of the same direction are studied using quantum mechanical methods.We derive analytical equations for the ionization probability in dependence on characteristic parameters of the system for both extreme cases of a constan telectric field and of a linearly polarized electromagnetic wave.The ionization probabilities for a superposition of dc and low frequency ac electric fields of the same direction are calculated.The impulse distribution of ionization probability for a system bound by short range forces is found for a superposition of constant and alternating fields. The total probability for this process per unit of timeis derived within exponential accuracy.Forthe first time the influence of alternating electric field on electron tunneling probability induced by an electrostatic field is studied taking into account the pre-exponential term.

The authors present the approaches to portfolio creation as a strategy for professional development of a student-teacher, his goals, value paradigms and the range of possible structures.

The manual is devoted to the mathematical theory and methods of optimization applied to administrative decisions in economy. Volume 1 described approaches to mathematical modeling of management problems in economy and methods of mathematical programming tasks solution. Besides strict mathematical proofs, there are directing reasons, which is sometimes enough for understanding. There are many economic examples and exercises with detailed solutions. Readers are supposed to know the bases of the mathematical analysis and linear algebra, though necessary data from these courses in a concise form are provided in appendices.

A new statistical approach to alignment (finding the longest common subsequence) of two random RNA-type sequences is proposed. We have constructed a generalized ‘dynamic programming’ algorithm for finding the extreme value of the free energy of two noncoding RNAs. In our procedure, we take into account the binding free energy of two random heteropolymer chains which are capable of forming the cloverleaf-like spatial structures typical for RNA molecules. The algorithm is based on two observations: (i) the standard alignment problem can be considered as a zero-temperature limit of a more general statistical problem of binding of two associating heteropolymer chains; (ii) this last problem can be generalized naturally to consider sequences with hierarchical cloverleaf-like structures (i.e. of RNA type). The approach also permits us to perform a ‘secondary structure recovery’. Namely, we can predict the optimal secondary structures of interacting RNAs in a zero-temperature limit knowing only their primary sequences.