Growing perfect cubes

AbstractAn (n,a,b)-perfect double cube is a b×b×b sized n-ary periodic array containing all possible a×a×a sized n-ary array exactly once as subarray. A growing cube is an array whose cj×cj×cj sized prefix is an (nj,a,cj)-perfect double cube for j=1,2,…, where cj=njv/3,v=a3 and n1<n2<⋯. We construct the smallest possible perfect double cube (a 256×256×256 sized 8-ary array) and growing cubes for any a

Space-time segmentation method for study of the vertical structure and evolution of solar supergranulation from data provided by local helioseismology

Solar supergranulation remains a mystery in spite of decades of intensive studies. Most of the papers about supergranulation deal with its surface properties. Local helioseismology provides an opportunity to look below the surface and see the vertical structure of this convective structure. We present a concept of a (3+1)-D segmentation algorithm capable of recognising individual supergranules in a sequence of helioseismic 3-D flow maps. As an example, we applied this method to the state-of-the-art data and derived descriptive statistical properties of segmented supergranules -- typical size of 20--30 Mm, characteristic lifetime of 18.7 hours, and estimated depth of 15--20 Mm. We present preliminary results obtained on the topic of the three-dimensional structure and evolution of supergranulation. The method has a great potential in analysing the better data expected from the helioseismic inversions, which are being developed.Comment: 6 pages, 4 figures, accepted in New Astronom

Measurable circle squaring

Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third problem.Comment: 40 pages; Lemma 4.4 improved & more details added; accepted by Annals of Mathematic

Anomalous diffusion and response in branched systems: a simple analysis

We revisit the diffusion properties and the mean drift induced by an external field of a random walk process in a class of branched structures, as the comb lattice and the linear chains of plaquettes. A simple treatment based on scaling arguments is able to predict the correct anomalous regime for different topologies. In addition, we show that even in the presence of anomalous diffusion, Einstein's relation still holds, implying a proportionality between the mean square displacement of the unperturbed systems and the drift induced by an external forcing.Comment: revtex.4-1, 16 pages, 7 figure