Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs

We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #P hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as graph embeddings, discrete Morse theory and graph clustering.

Opto-fluidically multiplexed assembly and micro-robotics

Techniques for high-definition micromanipulations,such as optical tweezers,hold substantial interest across a wide range of disciplines.However,their applicability remains constrained by material properties and laser exposure.And while microfluidic manipulations have been suggested as an alternative,their inherent capabilities are limited and further hindered by practical challenges of implementation and control.Here we show that the iterative application of laser-induced,localized flow fields can be used for the relative positioning of multiple micro-particles,irrespectively of their material properties.Compared to the standing theoretical proposal,our method keeps particles mobile,and we show that their precision manipulation is non-linearly accelerated via the multiplexing of temperature stimuli below the heat diffusion limit.The resulting flow fields are topologically rich and mathematically predictable.They represent unprecedented microfluidic control capabilities that are illustrated by the actuation of humanoid micro-robots with up to 30 degrees of freedom,whose motions are sufficiently well-defined to reliably communicate personal characteristics such as gender,happiness and nervousness.Our results constitute high-definition micro-fluidic manipulations with transformative potential for assembly,micro-manufacturing,the life sciences,robotics and optohydraulically actuated micro-factories.

Active Nematodynamics on Curved Surfaces–The Influence of Geometric Forces on Motion Patterns of Topological Defects

We derive and numerically solve a surface active nematodynamics model.We validate the numerical approach on a sphere and analyse the influence of hydro-dynamics on the oscillatory motion of topological defects.For ellipsoidal surfaces the influence of geometric forces on these motion patterns is addressed by taking into ac-count the effects of intrinsic as well as extrinsic curvature contributions.The numerical experiments demonstrate the stronger coupling with geometric properties if extrinsic curvature contributions are present and provide a possibility to tuneflow and defect motion by surface properties.

ISO-FLUCS:symmetrization of optofluidic manipulations in quasi-isothermal micro-environments

Recently,it has been demonstrated that thermoviscous flows can be used for a range of fine micromanipulations,such as moving the cytoplasm of cells and developing embryos,intracellular rheology,and femtonewton-range force measurements.These flows,also known as focused-light-induced cytoplasmic streaming(FLUCS),are induced by mid-infrared laser scanning of a temperature spot through the sample.However,localized laser scanning can inflict temperature perturbations of several Kelvins on the sample,potentially eliciting unspecific biological responses.In this study,we demonstrate how exploiting symmetry relations during laser scanning effectively disentangles laser heating and flow induction.We introduce flow-neutral scan sequences that use dynamic photothermal stimuli and spatiotemporal symmetry relations of scanning bridging up to three distinct time scales.We leverage further insights from a recently published analytical model of flow fields to present quasi-homogenous temperature distributions that leave flow lines and their local and directed character largely invariant.We present practical,intuitive solutions through predesigned sets of scan lines with near isothermal distributions and demonstrate that they are sufficient to generate and control flows in Caenorhabditis elegans embryos on a magnitude well in excess of endogenous flow velocities.Our results enable the separation of two previously tightly linked classes of physical stimuli,introduce a new,even less invasive standard for performing FLUCS perturbations,and pave the way for new unexplored avenues in the fields of soft matter and biomedicine.