Verbal functions of a group

Daniele Toller, "Verbal functions of a group", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 46 (2014), pp.71-99The aim of this paper is the study of elementary algebraic subsets of a group G, first defined by Markov in 1944 as the solution-set of a one-variable equation over G. We introduce the group of words over G, and the notion of verbal function of G in order to better describe the family of elementary algebraic subsets. The intersections of finite unions of elementary algebraic subsets are called algebraic subsets of G, and form the family of closed sets of the Zariski topology \Zar_G on G. Considering only some elementary algebraic subsets, one can similarly introduce easier-to-deal-with topologies \mathfrak T \subseteq \Zar_G, that nicely approximate \Zar_G and often coincide with it

Metrizability of hereditarily normal compact like groups

Inspired by the fact that a compact topological group is hereditarily normal if and only if it is metrizable, we prove that various levels of compactness-like properties imposed on a topological group G allow one to establish that G is hereditarily normal if and only if G is metrizable (among these properties are locally compactness, local minimality and \omega-boundedness). This extends recent results from [4] in the case of countable compactness

Algebraic entropy on strongly compactly covered groups

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact open normal subgroup of G. For continuous endomorphisms \u3c6:G\u2192G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions

The algebraic entropy of one-dimensional finitary linear cellular automata

The aim of this paper is to present one-dimensional finitary linear cellular automata $S$ on $\mathbb Z_m$ from an algebraic point of view. Among various other results, we: (i) show that the Pontryagin dual $\widehat S$ of $S$ is a classical one-dimensional linear cellular automaton $T$ on $\mathbb Z_m$; (ii) give several equivalent conditions for $S$ to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of $S$, which coincides with the topological entropy of $T=\widehat S$ by the so-called Bridge Theorem. In order to better understand and describe the entropy we introduce the degree $\mathrm{deg}(S)$ and $\mathrm{deg}(T)$ of $S$ and $T$.Comment: 21 page

Optimality-preserving Reduction of Chemical Reaction Networks

Across many disciplines, chemical reaction networks (CRNs) are an established population model defined as a system of coupled nonlinear ordinary differential equations. In many applications, for example, in systems biology and epidemiology, CRN parameters such as the kinetic reaction rates can be used as control inputs to steer the system toward a given target. Unfortunately, the resulting optimal control problem is nonlinear, therefore, computationally very challenging. We address this issue by introducing an optimality-preserving reduction algorithm for CRNs. The algorithm partitions the original state variables into a reduced set of macro-variables for which one can define a reduced optimal control problem from which one can exactly recover the solution of the original control problem. Notably, the reduction algorithm runs with polynomial time complexity in the size of the CRN. We use this result to reduce reachability and control problems of large-scale protein-interaction networks and vaccination models with hundreds of thousands of state variables