Narrow Results

For a finite player cooperative cost game, we consider two solutions that are based on excesses of coalitions. We define per-capita excess-sum of a player as sum of normalized excesses of coalitions involving this player and view it as a measure of player's dissatisfaction. So, per-capita excess-sum allocation is that imputation that minimizes the maximum per-capita excess-sums of players. We provide a closed form expression for an allocation, which is the per-capita excess-sum allocation if it is also individually rational. We propose a finite step algorithm to compute per-capita excess-sum allocation for a general game. We show that per-capita excess-sum allocation is coalitionally monotonic. Next, we consider excess-sum solution wherein a player views entire coalition's excess as a measure of dissatisfaction. This excess-sum solution also has above properties. In addition, we consider a super set of core and show that excess-sum allocation can be viewed as an imputation that is a certain center of this polyhedron. We introduce a class of cooperative games that can model cost sharing among divisions of a firm when they buy items at volume discounts. We characterize when excess-based allocations coincide with Shapley value, nucleolus, etc. in such games.

In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\ge 2$. We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc.33 (1931) 263–321] and Rado [On Plateau’s problem, Ann. Math.31 (1930) 457–469]. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-$g$ minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding–Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory by Colding–Minicozzi and the author to all genera.

This is a brief introduction to the Classic on Constructive Game Theory by Max Euwe (18), translated and republished in this issue of New Mathematics and Natural Computation. The task and aim of the Introduction is simply to provide a context, and some historical notes, particularly on the difference between Brouwer's approach to Set Theory — an Intuitionistic-Constructive approach and that of Zermelo.

In this paper, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, a complete Möbius band or a complete Riemannian plane leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.