Algorithms for ham-sandwich cuts

Given disjoint setsP ₁,P ₂, ...,P _d inR ^d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR ^d−1 . With the current estimates, we get complexity close toO(n ^3/2 ) ford=3, roughlyO(n ^8/3 ) ford=4, andO(n ^d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR ³ when the three sets are suitably separated. A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Symmetries of Integrable Difference Equations on the Quad-Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.     

Trace norm estimates for products of integral operators and diffusion semigroups

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.