"Our result may be generalized to other domains where similar algorithms may apply. Our algorithm is based on the fact that the eigenfunctions are presented as a linear combination of simple plane waves. It is therefore tempting to try and generalize it for other drums with similar property. The equilateral triangle is an immediate candidate (see [29] and references within).

A further, and quite surprising, result is the recursive formula for the number of nodal loops. To our knowledge this is the first known exact formula for the nodal count of a non-separable planar manifold (for certain eigenfunctions of tori exact formulas have been given in [22]). The formula was found by direct inspection of large tables and has been verified for a large bulk of data computationally. An obvious challenge is to prove this formula. In particular, the recursive part of the formula resembles the famous Euclid algorithm for the greatest common divisor. A further investigation of the mentioned formula might therefore expose some new number theoretical properties of the nodal count."

"Our 2.546-approximation is quite simple. The performance guarantee is based on a simple area argument. This gives rise to the following question: what is the smallest square that suffices for packing any set of circles of total area 1? We believe the worst-case may very well be shown in Figure 13, which yields a lower bound of 1.471299... We believe there are relatively easy ways to improve the upper bound."

"We believe that our result could shed the light on the relation between the distribution of motifs and the struc- ture of the adjacency matrix of a hierarchical network. However to make this relation more profound the “in- verse” problem should be considered as well. Namely, it would be desirable to check if the stable distribution of motifs is uniquely related to any kind of hierarchical organization of the network."

"A variety of open questions and extensions remain to be studied. It may be interesting to consider the algorithmic question of computing, for a given configuration of an adorned chain (whose adornments are not slender), whether or not there exists a motion that opens it.

Our results have application to hinged dissections of polyregulars, e.g., polyominoes; this in- cludes the (slender) case of squares connected at opposite corners. An interesting question arising in this context is whether every hinged dissection can be subdivided so that the pieces are slender."

"The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916 [23]; see also [25], [2]. Let C be a convex body in the plane. Following Bagemihl [2], we call a set B an opaque set or a barrier for C, if it meets all lines that intersect C. A barrier may consist of one or more rectifiable arcs. It does not need to be connected and its portions may lie anywhere in the plane, including the exterior of C; see [2], [4].

What is the length of the shortest barrier for a given convex body C? In spite of considerable efforts, the answer to this question is not known even for the simplest instances of C, such as a square, a disk, or an equilateral triangle…"

"…We can also consider what happens when instead of only considering a single type of shuffling we consider combining the j! different shuffling rules that come from all the possible rearrangements of the ordering of the labels. And of course, perhaps the most important thing missing right now is a good magic trick that can be performed using this shuffling rule, which was the original motivation of Larry Carter and J.-C. Reyes who first suggested this problem!"

"We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. "

"Every human likes choices. But today's fast route planning algorithms usually compute just a single route between source and target. There are beginnings to compute alternative routes, but this topic has not been studied thoroughly. Often, the aspect of meaningful alternative routes is neglected from a human point of view. We fill in this gap by suggesting mathematical definitions for such routes. As a second contribution we propose heuristics to compute them, as this is NP-hard in general."