Related papers: 1D Triple Bubble Problem with Log-Convex Density

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex.

We derive boundary conditions at interfaces (contact discontinuities) for a class of Lagrangian models describing, in particular, bubbly flows. We use these conditions to study Kelvin-Helmholtz' instability which develops in the flow of two…

The properties of the Local Bubble, Local Fluff complex of nearby interstellar clouds, and the heliosphere are mutually constrained by data and theory. Observations and models of the diffuse radiation field, interstellar ionization, pick-up…

Let A \subseteq [1,..,N]^2 be a set of density at least 1/(log log log N)^c, where c some constant c>0. We prove that A contains a so-called right-angle triangle, i.e. a triple of the form {(k,m), (k+d,m), (k,m+d)}, where d>0.

We prove that the maximal number of conics, a priori irreducible of reducible, on a smooth spatial quartic surface is 800, realized by a unique quartic. We also classify quartics with many (at least 720) conics. The maximal number of real…

We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of…

We present a unified numerical method to determine the shapes of multiple Hele-Shaw bubbles in steady motion, and in the absence of surface tension, in three planar domains: free space, the upper half-plane, and an infinite channel. Our…

This article continues the research initiated in [17]. We derive explicit formulae for the leading order profiles of eleven types of stationary solutions to a one-dimensional two-layer thin-film liquid model considered with an…

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…

Since the thesis of K. Reinhardt in 1918, it is well known that there are exactly three types of convex hexagons that can tile the plane. However, the proof of the fact is far from being complete. We prove this fact, under an assumption…

We give a topological bound on the number of minimal models of a class of three dimensional log smooth pairs of general type.

Bubbles are point-like regular solutions of the higher-dimensional Kaluza-Klein equations that appear as naked singularities in four dimensions. We analyze all such possible solutions in 5D Kaluza-Klein theory that are static and…

We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these…

For a smooth, projective family of homogeneous varieties defined over a number field, we show that if potential density holds for the rational points of the base, then it also holds for the total space. A conjecture of Campana and…

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically $12^n…

We construct a family of smooth charged bubbling solitons in $\mathbb{M}^4 \times$T$^2$, four-dimensional Minkowski with a two-torus. The solitons are characterized by a degeneration pattern of the torus along a line in $\mathbb{M}^4$…

Recent work of Ballas, Cooper, and Leitner identifies $(n+1)$ types of $n$-dimensional convex projective cusps, one of which is the standard hyperbolic cusp. Work of Ballas-Marquis, and Ballas-Danciger-Lee give examples of these exotic…

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we…

Experiments are conducted to study the path and shape of single air bubbles (diameter range 0.10- 0.20cm) rising freely in clean water. The experimental results demonstrate that the bubble shape has a bistable state, i. e. the bubble…

We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the…