Cluster algebras of infinite rank as colimits

We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism without specializations. Assem, Dupont and Schiffler proposed the problem of a classification of ideal rooted cluster morphisms. We provide a partial solution by showing that every rooted cluster morphism without specializations is ideal, but in general rooted cluster morphisms are not ideal.Comment: Included cluster algebras of uncountable rank, fixed some typos. Results on the countable case unchanged, comments appreciate

Mutation of torsion pairs in cluster categories of Dynkin type DD

Mutation of torsion pairs in triangulated categories and its combinatorial interpretation for the cluster category of Dynkin type $A_n$ and of type $A_\infty$ have been studied by Zhou and Zhu. In this paper we present a combinatorial model for mutation of torsion pairs in the cluster category of Dynkin type $D_n$, using Ptolemy diagrams of Dynkin type $D_n$ which were introduced by Holm, J{\o}rgensen and Rubey.Comment: Corrected typos, some arguments made more concise, results unchange

Homotopy invariants of singularity categories

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.Comment: final revisio

Cluster tilting modules for mesh algebras

We study cluster tilting modules in mesh algebras of Dynkin type, providing a new proof for their existence. In all but one case, we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics.Comment: comments appreciated; the third version includes a discussion on the combinatorics of the mutation

Finite Time Robust Control of the Sit-to-Stand Movement for Powered Lower Limb Orthoses

This study presents a technique to safely control the Sit-to-Stand movement of powered lower limb orthoses in the presence of parameter uncertainty. The weight matrices used to calculate the finite time horizon linear-quadratic regulator (LQR) gain in the feedback loop are chosen from a pool of candidates as to minimize a robust performance metric involving induced gains that measure the deviation of variables of interest in a linear time-varying (LTV) system, at specific times within a finite horizon, caused by a perturbation signal modeling the variation of the parameters. Two relevant Sit-to-Stand movements are simulated for drawing comparisons with the results documented in a previous work.Comment: 8 pages, 14 figures, ACC 2018 Submissio

Non-linear estimation is easy

Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with on-line estimations, are illustrating our viewpoint