Related papers: A note on the pyjama problem
We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in…
Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming…
A set of n segments in the plane may form a Euclidean TSP tour, a tree, or a matching, among others. Optimal TSP tours as well as minimum spanning trees and perfect matchings have no crossing segments, but several heuristics and…
Given a set of disjoint simple polygons $\sigma_1, \ldots, \sigma_n$, of total complexity $N$, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by…
We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet, and identify the $\Gamma$-limit as the sheet thickness goes to 0, thus extending the previous work of the first author [Bella,…
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have…
For all $k \geq 1$, we show that deciding whether a graph is $k$-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is…
By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solution of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime…
We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique…
We consider harmonic maps from Minkowski space into the three sphere. We are especially interested in solutions which are asymptotically constant, i.e. converge to the same value in all directions of spatial infinity. Physical 3-space can…
In this paper we apply a geometric covering method to study the number of ends on shrinkers. On one hand, we prove that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree. On the other…
Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an…
Based on fundamental properties of light scattering by a particle we reveal the existence of the ultimate upper limit for the light absorption by any partial mode. First, we obtain this result for scattering of a plane wave by a symmetric…
Let P be a polygon with rational vertices in the plane. We show that for any finite odd-sized collection of translates of P, the area of the set of points lying in an odd number of these translates is bounded away from 0 by a constant…
We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem.
Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two…
Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which overlaps of certain specified sizes are forbidden.…
Alon and F\"{u}redi (1993) showed that the number of hyperplanes required to cover $\{0,1\}^n\setminus \{0\}$ without covering $0$ is $n$. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the…
In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $\sigma=1/2,$ when the prescribing $\sigma$-curvature function satisfies the…
Given a noncompact disconnected complete periodic curve $\Gamma$ with no self intersection in $\mathbb R^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $\Gamma$. As an application it is…