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Fedor Duzhin

Publications and source records attributed to Fedor Duzhin.

4 recordsLinked to original sources

Learning in teams: peer evaluation for fair assessment of individual contributions

The "free rider" problem has long plagued pedagogies based on collaborative learning. The most common solution to the free rider problem is peer evaluation. As well other existing methods of peer evaluation include self-evaluation --- and hence are prone to grade inflation or, as we show here, are inaccurate in that they do not fairly reward the most hard working student. Another common concern with existing methods of peer evaluation is that students often do not have the necessary skills to evaluate the work of their peers objectively. In this paper, we introduce a new mechanism for peer evaluation that does not rely on self-evaluation, and yet remains accurate, i.e., if all students are completely truthful in their evaluations, then the output of our mechanism becomes an objective truth. At the same time, our mechanism integrates the instructor's judgment with respect to the credibility of students' evaluations. For example, the instructor gives scores to students for writing credible reviews and, in turn, subsequent students' evaluations are weighted according to these instructor scores.

math.HO

Lower bounds on the number of closed trajectories of generalized billiards

Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from the boundary making the angle of incidence equal the angle of reflection. The billiard motion can happen to be a closed (or periodic) one when the billiard ball rebounds k times and then gets to the initial position with the same speed vector as in the beginning. The study of closed billiard trajectories is due to George Birkhoff who in 1927 proved a lower estimate for the number of closed billiard trajectories of a certain period k. We consider the most general case when the billiard ball reflects from an arbitrary submanifold of a Euclidean space. We prove Morse inequalities in this situation and apply them to find a lower estimate for the number of closed billiard trajectories of any prime period in terms of Betti numbers of the given manifold.

math.DG