Abstract. We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of factor maps between the case of continuous and non-continuous weight functions.

Model sets are projections of certain lattice subsets. It was realised by Moody that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map which associates lattice subsets to points of the torus and then transferring the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrisation map, and we derive a formula for its pattern frequencies. * These notes profited enormously from talks by and/or discussions with Michael Baake, Tobias Hartnick, Johannes Kellendonk, Mariusz Lemańczyk, Daniel Lenz, Tobias Oertel-Jäger and Nicolae Strungaru, from several inspiring workshops in the framework of the DFG Scientific Network "Skew Product Dynamics and Multifractal Analysis" organised by Tobias Oertel-Jäger, and finally from very helpful and supporting comments of an anonymous referee.• we obtain as a warm-up some relatives to well known topological results on maximal equicontinuous factors and the lack of weak mixing (Theorem 1),• we can introduce a general notion of Mirsky measures, namely the push forward of Haar measure onX under the map ν W , compare [19,17,34,51],• we show that the systems equipped with this Mirsky measure have pure point dynamical spectrum (Theorem 2),• we prove strict ergodicity when m H (∂W) = 0 (Theorem 2),• we identify the Mirsky measure as the unique invariant measure with maximal density for typical configurations (Theorem 4),• we show that the configurations with maximal density are precisely the generic points for the Mirsky measure (Theorem 5),• and we deduce from this a formula for the pattern frequencies of configurations with maximal density (Remark 3.12), which is also discussed in [5, Rem. 5].While the measure theoretic assertions from this list "survive" the passage to configurations on G even if the relevant projection is not 1-1, some finer information can be transferred in this way only if that projection is 1-1 when restricted to sufficiently large subsets. This issue is discussed in Section 4 for model sets with interval windows, topologically regular windows and B-free systems.

We analyze new data for self-avoiding polygons, on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted self-avoiding polygons may satisfy a q-algebraic functional equation.

Abstract. We prove that the diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. This is achieved using Fourier analysis of unbounded measures on locally compact abelian groups, as developed by Argabright and de Lamadrid. We also discuss related diffraction results for certain classes of non-regular so-called weak model sets.

Poland-Scheraga models were introduced to describe the DNA denaturation transition. We give a rigorous and refined discussion of a family of these models. We derive possible scaling functions in the neighborhood of the phase transition point and review common examples. We introduce a self-avoiding Poland-Scheraga model displaying a first order phase transition in two and three dimensions. We also discuss exactly solvable directed examples. This complements recent suggestions as to how the Poland-Scheraga class might be extended in order to display a first order transition, which is observed experimentally.first order phase transition would be the appropriate description. The question whether such an asymptotic description, which implies very long chains, is valid for relatively short DNA sequences, has been discussed recently [16,22]. Nevertheless, a directed extension of the PS model, being essentially a one-dimensional Ising model with statistical weighting factors for internal loops, is widely used today and yields good coincidence of simulated melting curves with experimental curves for known DNA sequences [35,4,5]. Another recent application of PS models analyses the role of mismatches in DNA denaturation [13]. A numerical approach to DNA denaturation, which we will not discuss further, uses variants of the Peyrard-Bishop model [29,34], a Hamiltonian model of two harmonic chains coupled by a Morse potential.With the advent of efficient computers, it has more recently been possible to simulate analytically intractable models extending the PS class, which are assumed to be more realistic representations of the biological problem. One of these is a model of two self-avoiding and mutually avoiding walks, with an attractive interaction between different walks at corresponding positions in each walk [9,8,3,1]. The model exhibits a first order phase transition in d = 2 and d = 3. The critical properties of the model are described by an exponent c ′ related to the loop length distribution [20,21,23,8,3,1], see also Fisher's review article [11]. This exponent is called c again. Indeed, for PS models, it coincides with the loop class exponent c if 1 < c < 2, see below. Within a refined model, where different binding energies for base pairs and stiffness are taken into account, the exponent c ′ seems to be largely independent of the specific DNA sequence and of the stiffness of paired walk segments corresponding to double stranded DNA parts [8]. There are, however, no simulations of melting curves for known DNA sequences which are compared to experimental curves for this model.An approximate analytic derivation of the exponent related to this new model was given by Kafri et al. [20,21,23] using the theory of polymer networks. They estimated the excluded volume effect arising from the interaction between a single loop and two attached walks. This approach (refined recently [3]) yields an approximation to the loop length distribution exponent c ′ , which agrees well with simulation results of interacting self-avoiding wa...