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The Spectre is a family of recently discovered aperiodic monotiles that tile the plane only in non-periodic ways, and novel physical phenomena have been predicted for planar systems made of aperiodic monotiles. It is shown that point…
The static diffraction intensity distribution from large material system conceived as perfectly homogeneous system made inhomogeneous, though substitution of groups of atoms, small particles, by other groups of atoms, is explicitly…
The scaling behaviour of the diffraction intensity near the origin is investigated for (partially) ordered systems, with an emphasis on illustrative, rigorous results. This is an established method to detect and quantify the fluctuation…
We focus on open questions regarding the uniqueness of distributional solutions of the fast diffusion equation (FDE) with a given source term. When the source is sufficiently smooth, the uniqueness follows from standard results. Assuming…
Discrete diffusion models are a new class of text generators that offer advantages such as bidirectional context use, parallelizable generation, and flexible prompting compared to autoregressive models. However, a critical limitation of…
Discontinuous transitions into absorbing states require an effective mechanism that prevents the stabilization of low density states. They can be found in different systems, such as lattice models or stochastic differential equations (e.g.…
The diffraction of fast atoms at crystal surfaces is ideal for a detailed investigation of the surface electronic density. However, instead of sharp diffraction spots, most experiments show elongated streaks characteristic of inelastic…
Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead…
Percolation is a paradigmatic model in disordered systems and has been applied to various natural phenomena. The percolation transition is known as one of the most robust continuous transitions. However, recent extensive studies have…
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there…
We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be understood within the framework of unitary group representations. This allows us to prove an orthogonality feature of measures…
We review and study the correspondence between discrete linear lattice/chain models of interacting particles and their continuous counterparts represented by linear partial differential equations. In particular, we study the correspondence…
The diffraction spectrum of the dart-rhombus random tiling of the plane is derived in rigorous terms. Using the theory of dimer models, it is shown that it consists of Bragg peaks and an absolutely continuous diffuse background, but no…
New aspects of a relation between lattice and dislocation structures are examined within a physically transparent theoretical scheme. Predicted features originating from the lattice discreteness include: (i) multiple core dislocation…
We present a simplified model for dynamical diffraction of particles through a periodic thick perfect crystal based on repeated application of a coherent beam splitting unitary at coarse-grained lattice sites. By demanding translational…
We consider discrete-time switching systems composed of a finite family of affine sub-dynamics. First, we recall existing results and present further analysis on the stability problem, the existence and characterization of compact…
We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map…
Flatness of discrete-time systems can be characterized by two simple properties. There exists a map, a submersion, from the flat coordinates and their forward shifts to the state and the input of the discrete-time system, such that the…
Recent progress in text-to-image (TTI) systems, such as StableDiffusion, Imagen, and DALL-E 2, have made it possible to create realistic images with simple text prompts. It is tempting to use these systems to eliminate the manual task of…
Unlike the (classical) Kolakoski sequence on the alphabet {1,2}, its analogue on {1,3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic…