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.

At the age of almost three quarters of a century, quantum mechanics is by all accounts a mature theory. There were times when it seemed that it had borne its best fruit already and would give way to investigation of deeper levels of matter. Today this sounds like rash thinking. Modern experimental techniques have led to discoveries of numerous new quantum effects in solid state, optics and elsewhere. Quantum mechanics is thus gradually becoming a basis for many branches of applied physics, in this way entering our everyday life. While the dynamic laws of quantum mechanics are well known, a proper theoretical understanding requires methods which would allow us to de rive the abundance of observed quantum effects from the first principles. In many cases the rich structure hidden in the Schr6dinger equation can be revealed only using sophisticated tools. This constitutes a motivation to investigate rigorous methods which yield mathematically well-founded properties of quantum systems.

We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.

We consider a family \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} of \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} -periodic Schrdinger operators with \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} -interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} surfaces. We show that in the limit when \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} and the interactions strengths are appropriately scaled, \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} has at most \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} gaps within finite intervals, and moreover, the limiting behavior of the first \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

We investigate spectral properties of the Laplacian in <i>L</i>² (<i>Q</i>), where <i>Q</i> is a tubular region in ³ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.