Investigating self-similar groups using their finite LL-presentation

Self-similar groups provide a rich source of groups with interesting properties; e.g., infinite torsion groups (Burnside groups) and groups with an intermediate word growth. Various self-similar groups can be described by a recursive (possibly infinite) presentation, a so-called finite $L$-presentation. Finite $L$-presentations allow numerous algorithms for finitely presented groups to be generalized to this special class of recursive presentations. We give an overview of the algorithms for finitely $L$-presented groups. As applications, we demonstrate how their implementation in a computer algebra system allows us to study explicit examples of self-similar groups including the Fabrykowski-Gupta groups. Our experiments yield detailed insight into the structure of these groups

What Makes a Muslim Intellectual? On the Pro's and Con's of a Category

<p style="margin-right: 1cm; margin-bottom: 0cm; line-height: 100%;" lang="en-GB">At its core, this essay contains a substantiated plea for bringing about conceptual clarity to the notion of “Muslim intellectual”, which the frequent and highly ideologically charged public usage of this term seems to distort. In search for a sound analytical concept of “intellectual” first, relevant sociological and philosophical deliberations are highlighted, indicating that both of their notions differ to such an extent that their applicability to academic pursuit must be doubted. Yet, by discussing some considerations by a Study of Islam open to the approaches of the Social Sciences a possible framework for an analytically meaningful concept of “Muslim intellectual” is presented. At the same time, however, arguments are presented for why those contemporary Muslim thinkers who are usually credited with being “Muslim intellectuals” would hardly fit the analytical criteria for such label.</p

Zeta-regularized vacuum expectation values

It has recently been shown that vacuum expectation values and Feynman path integrals can be regularized using Fourier integral operator $\zeta$-function, yet the physical meaning of these $\zeta$-regularized objects was unknown. Here we show that $\zeta$-regularized vacuum expectations appear as continuum limits using a certain discretization scheme. Furthermore, we study the rate of convergence for the discretization scheme using the example of a one-dimensional hydrogen atom in $(-\pi,\pi)$ which we evaluate classically, using the Rigetti Quantum Virtual Machine, and on the Rigetti 8Q quantum chip "Agave" device. We also provide the free radiation field as an example for the computation of $\zeta$-regularized vacuum expectation values in a gauge theory.Comment: 36 pages, 2 figures; accepted version (Journal of Mathematical Physics