Bounds on the global dimension of certain piecewise hereditary categories

We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables. We use this to show that the global dimension of a finite dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.Comment: 7 pages; slightly revised; to appear in J. Pure Appl. Algebr

On Galois coverings and tilting modules

Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence between the Galois coverings with group G of A and the Galois coverings with group G of B. The hypothesis on T is expressed using the Hasse diagram of basic tilting A-modules and is always verified if A is of finite representation type. Then, we use the above correspondence to prove that A is simply connected if and only if B is simply connected, under the same hypothesis on T. Finally, we prove that if a tilted algebra B of type Q is simply connected, then Q is a tree and the first Hochschild cohomology group of B vanishesComment: Fourth version. A result on the simple connectedness of tilted algebras was adde

Tame concealed algebras and cluster quivers of minimal infinite type

In this paper we explain how and why the list of Happel-Vossieck of tame concealed algebras is closely related to the list of A. Seven of minimal infinite cluster quivers.Comment: 16 pages, new version with an additional section on cluster-tilted algebras of minimal infinite typ

Passive swimming in low Reynolds number flows

The possibility of microscopic swimming by extraction of energy from an external flow is discussed, focusing on the migration of a simple trimer across a linear shear flow. The geometric properties of swimming, together with the possible generalization to the case of a vesicle, are analyzed.The mechanism of energy extraction from the flow appears to be the generalization to a discrete swimmer of the tank-treading regime of a vesicle. The swimmer takes advantage of the external flow by both extracting energy for swimming and "sailing" through it. The migration velocity is found to scale linearly in the stroke amplitude, and not quadratically as in a quiescent fluid. This effect turns out to be connected with the non-applicability of the scallop theorem in the presence of external flow fields.Comment: 4 pages, 4 figure

Efficiency of surface-driven motion: nano-swimmers beat micro-swimmers

Surface interactions provide a class of mechanisms which can be employed for propulsion of micro- and nanometer sized particles. We investigate the related efficiency of externally and self-propelled swimmers. A general scaling relation is derived showing that only swimmers whose size is comparable to, or smaller than, the interaction range can have appreciable efficiency. An upper bound for efficiency at maximum power is 1/2. Numerical calculations for the case of diffusiophoresis are found to be in good agreement with analytical expressions for the efficiency

Pair diffusion, hydrodynamic interactions, and available volume in dense fluids

We calculate the pair diffusion coefficient D(r) as a function of the distance r between two hard-sphere particles in a dense monodisperse suspension. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine D(r) from the propagators (Green's functions) of particle pairs obtained from discontinuous molecular dynamics simulations. At distances exceeding 3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamic theory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/r distance dependence of D(r) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated D(r) from the hydrodynamic models at short distances r reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry.Comment: 6 pages, 5 figure

Tank-treading as a means of propulsion in viscous shear flows

The use of tank-treading as a means of propulsion for microswimmers in viscous shear flows is taken into exam. We discuss the possibility that a vesicle be able to control the drift in an external shear flow, by varying locally the bending rigidity of its own membrane. By analytical calculation in the quasi-spherical limit, the stationary shape and the orientation of the tank-treading vesicle in the external flow, are determined, working to lowest order in the membrane inhomogeneity. The membrane inhomogeneity acts in the shape evolution equation as an additional force term, that can be used to balance the effect of the hydrodynamic stresses, thus allowing the vesicle to assume shapes and orientations that would otherwise be forbidden. The vesicle shapes and orientations required for migration transverse to the flow, together with the bending rigidity profiles that would lead to such shapes and orientations, are determined. A simple model is presented, in which a vesicle is able to migrate up or down the gradient of a concentration field, by stiffening or softening of its membrane, in response to the variations in the concentration level experienced during tank-treading.Comment: 21 pages, 4 figure

Algebras of acyclic cluster type: tree type and type A~\widetilde{A}

In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type \A_n for each possible orientation of \A_n. We give an explicit way to read off in which derived equivalence class such an algebra lies, and describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.Comment: v2: 37 pages. Title is changed. A mistake in the previous version is now corrected (see Remark 3.14). Other changes making the paper coherent with the version 2 of 1003.491