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The built-up land represents an important type of an overall landscape. In this paper, the built-up land structure in the largest cities in the Czech Republic and some selected cities in the USA is analysed using the framework of statistical physics. We calculate the variance of the total area and of the count of the built-up land plots contained inside discs of different radii. In both cases, the variance as a function of the disc radius follows a power law with exponents that are comparable through different cities. The study is based on the cadastral data from the Czech Republic and on the building footprints from GIS data in the USA.

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Micciancio defined a generalization of cyclic lattices, called ideal lattices. These lattices can be used in cryptosystems to decrease the number of parameters necessary to describe a lattice by a square root, making them more efficient. He proves that the computational intractability of classic lattice problems for these lattices gives rise to provably secure one-way and collision-resistant hash functions. This provable security relies on the assumption that reducing bases of ideal lattices is similar to reducing bases of random lattices. We give an indication that lattice problems in ideal lattices do not represent the general case by providing a distinguisher, which decides in time O(n^4) whether a given basis of rank spans an ideal lattice or not. Using this algorithm we perform a statistical analysis for several dimensions and show that randomly generated lattices are practically never ideal.