In this article, using the computer, are enumerated all locally-rigid packings by $N$ congruent circles (spherical caps) on the unit sphere ${Bbb S}^2 $ with $N < 12.$ This is equivalent to the enumeration of irreducible spherical contact graphs.
Recent studies on nanoscale field-effect sensors reveal the crucial importance of the low frequency noise for determining the ultimate detection limit. In this letter, the 1/f-type noise of Si nanoribbon field-effect sensors is investigated. We demon
strate that the signal-to-noise ratio can be increased by almost two orders of magnitude if the nanoribbon is operated in an optimal gate voltage range. In this case, the additional noise contribution from the contact regions is minimized, and an accuracy of 0.5% of a pH shift in one Hz bandwidth can be reached.
The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The pr
oblem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.
There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected edge unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of L).
