Olga Kosheleva

Towards Optimal Effort Distribution in Process Design under Uncertainty, with Application to Education

In most application areas, we need to take care of several (reasonably independent) participants. For example, in controlling economics, we must make sure that all the economic regions prosper. In controlling environment, we want to guarantee that all the geographic regions have healthy environment. In education, we want to make sure that all the students learn all the needed knowledge and skills. In real life, the amount of resources is limited, so we face the problem of “optimally” distributing these resources between different objects. What is a reasonable way to formalize “optimally”? For each of the participants, preferences can be described by utility functions: namely, an action is better if its expected utility is larger. It is natural to require that the resulting group preference has the following property: if two actions has the same quality for all participants, then they are equivalent for the group as well. It turns out that under this requirement, the utility function of a group is a linear combination of individual utility functions. To solve the resulting optimization problem, we need to know, for each participant i, the utility resulting from investing effort e in this participant. In practice, we only know this value with (interval) uncertainty. So, for each distribution of efforts, instead of a single value of the group utility, we only have an interval of possible values. To compare such intervals, we use Hurwicz optimism-pessimism criterion well justified in decision making. In the talk, we propose a solution to the resulting optimization problem.

Open-ended configurations of radio telescopes: towards optimal design

The quality of radio astronomical images drastically depends on where we place the radio telescopes. During the design of the very large array, it was empirically shown that the power law design, in which n-th antenna is placed at a distance n/sup /spl beta// from the center, leads to the best image quality. In this paper, we provide a theoretical justification for this empirical fact.

How to Make A Proof of Halting Problem More Convincing: A Pedagogical Remark

As an example of an algorithmically undecidable problem, most textbooks list the impossibility to check whether a given program halts on given data. A usual proof of this result is based on the assumption that the hypothetical halt-checker works for all programs. To show that a halt-checker is impossible, we design an auxiliary program for which the existence of such a halt-checker leads to a contradiction. However, this auxiliary program is usually very artificial. So, a natural question arises: what if we only require that the halt-checker work for reasonable programs? In this paper, we show that even with such a restriction, haltcheckers are not possible – and thus, we make a proof of halting problem more convincing for students. 1 Formulation of the Problem Halting problem: reminder. A computer science degree means acquiring both the practical skills needed to design and program software and the theoretical knowledge describing which computational tasks are possible and which are not. Different programs include different examples of problems for which no computational solution is possible, but all of them include – with proof – the very first example of such a problem: the halting problem, according to which no algorithm is possible that, given a program p and data d, always checks whether p halts on d; see, e.g., [2]. Some textbooks describe this result in terms of Turing machines but, in our opinion, this result is much clearer to students when it is described in terms of programs – i.e., something with which are very familiar – rather than in terms of Turing machines, a new concept that they have just learned in the corresponding theoretical course and with which they are not yet very familiar. Let us therefore concentrate on the formulation of this result in terms of programs.

Why Zipf's law: a symmetry-based explanation

Abstract In many practical situations, we have probability distributions for which, for large values of the corresponding quantity x, the probability density has the form ρ(x) ∼ x−α for some α > 0. While, in principle, we have laws corresponding to different α, most frequently, we encounter situations – first described by Zipf for linguistics – when α ≈ 1. The fact that Zipf’s has appeared frequently in many different situations seems to indicate that there must be some fundamental reason behind this law. In this paper, we provide a possible explanation.