We investigate the effects of the spatial discreteness of molecules in reaction-diffusion systems. It is found that discreteness within the so-called Kuramoto length can lead to a localization of molecules, resulting in novel steady states that do not exist in the continuous case. These states are analyzed theoretically as the fixed points of accelerated localized reactions, an approach that wa...

To study the fluctuations and dynamics in chemical reaction processes, stochastic differential equations based on the rate equation involving chemical concentrations are often adopted. When the number of molecules is very small, however, the discreteness in the number of molecules cannot be neglected since the number of molecules must be an integer. This discreteness can be important in biochem...

After sketching a context in which to seek observable signals of spatiotemporal discreteness, I briefly review the status of the causal set program for quantum gravity, concluding with a simple model for the field produced by a moving charge in a background causal set. 1. How might discreteness show up? Riemann had a nice phrase somewhere that talked about “the reality that underlies spacetime....

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichmüller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of such words and sequences: the semigroup of so called good words of Gehring-Martin...

The aim of this paper is to study silting-discrete triangulated categories. We establish a simple criterion for silting-discreteness in terms of 2-term silting objects. This gives a powerful tool to prove silting-discreteness of finite dimensional algebras. Moreover, we will show Bongartz-type Lemma for silting-discrete triangulated categories.

We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls...