The least-area hypersurface enclosing and separating two given volumes in R is the standard double bubble.

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R3.

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R3.

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic three-torus T 3. For comparable small volumes, we prove that an area minimizing double bubble in T 3 is the standard double bubble from R 3 .

The classical isoperimetric inequality in R states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120◦.

The classical isoperimetric inequality states that the surface of smallest area enclosing a given volume in R3 is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 2π/3.

In this paper, we show that the so-called “double bubbles” are not downward dense in the d.c.e. degrees. Here, a pair of d.c.e. degrees d1 > d2 > 0 forms a double bubble if all d.c.e. degrees below d1 are comparable with d2.

Shaker, a voltage-dependent K+ channel, is enriched in the mushroom bodies (MBs), the locus of olfactory learning in Drosophila. Mutations in the shaker locus are known to alter excitability, neurotransmitter release, synaptic plasticity, and olfactory learning. However, a direct link of Shaker channels to MB intrinsic neuron (MBN) physiology has not been documented. We found that transcripts f...

We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.