Packing problems, such as how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, it has only recently been proved that the face-centered cubic lattice has the highest possible packing fraction ϕ = π/ √ 18 ≈ 0.74. It is also well-known that certain random (amorphous) jammed packings have ϕ ≈ 0.64. Here we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely; up to ϕ = 0.68 − 0.71 for spheroids with an aspect ratio close to that of M&M'S r Candies, and even approach ϕ ≈ 0.74 for general ellipsoids. We suggest that the higher 1

We present an automated procedure to assign CATH and SCOP classifications to proteins whose FSSP score is available. CATH classification is assigned down to the topology level, and SCOP classification is assigned to the fold level. Because the FSSP database is updated weekly, this method makes it possible to update also CATH and SCOP with the same frequency. Our predictions have a nearly perfect success rate when ambiguous cases are discarded. These ambiguous cases are intrinsic in any protein structure classification that relies on structural information alone. Hence, we introduce the "twilight zone for structure classification." We further suggest that to resolve these ambiguous cases, other criteria of classification, based also on information about sequence and function, must be used.

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

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