A generic C1C^1 map has no absolutely continuous invariant probability measure

Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense $G_\delta) set in the$C^1$ topology. In the course of the proof, we need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result may be of independent interest.Comment: 12 page

Verslag workshops Friese Veenweidevisie

Waterschap en provincie werken aan een Veenweidevisie voor Friesland. In de Veenweidevisie vormt de zorg over de toenemende snelheid van de maaivelddaling en het verlies van de veenbodem een centraal aandachtspunt. Voor het verkennen van de problemen en oplossingen zijn streekbijeenkomsten georganiseerd. Drie scenario’s zijn voor de toekomst van de veenweiden in Friesland beschreven. Deze drie scenario’s zijn voor drie voorbeeldgebieden, Hommerts, Grote Veenpolder en het Buitenveld, uitgewerkt in drie workshops

Decoupling through input–output blending

The paper presents a novel decoupling method, based on blending the input and output signals of lineardynamical systems. For this purpose, blend vectors are introduced and calculated such that the mini-mum sensitivity of the controlled mode is maximised, while the worst case gain of the other subsystems is minimised from the blended input to the blended output. The problem is transformed to a standard optimisation program subject to Linear Matrix Inequality constraints. An arising rank constraint is resolved byan alternating projection scheme. The method is presented based on the decoupling of a single mode, but the extension to decouple multiple modes is also discussed. Numerical examples are given to validate the method and to illustrate how the proposed approach can be applied for control engineering problems

Predictive feedback control and Fitts' law

Fitts’ law is a well established empirical formula, known for encapsulating the “speed-accuracy trade-off”. For discrete, manual movements from a starting location to a target, Fitts’ law relates movement duration to the distance moved and target size. The widespread empirical success of the formula is suggestive of underlying principles of human movement control. There have been previous attempts to relate Fitts’ law to engineering-type control hypotheses and it has been shown that the law is exactly consistent with the closed-loop step-response of a time-delayed, first-order system. Assuming only the operation of closed-loop feedback, either continuous or intermittent, this paper asks whether such feedback should be predictive or not predictive to be consistent with Fitts law. Since Fitts’ law is equivalent to a time delay separated from a first-order system, known control theory implies that the controller must be predictive. A predictive controller moves the time-delay outside the feedback loop such that the closed-loop response can be separated into a time delay and rational function whereas a non- predictive controller retains a state delay within feedback loop which is not consistent with Fitts’ law. Using sufficient parameters, a high-order non-predictive controller could approximately reproduce Fitts’ law. However, such high-order, “non-parametric” controllers are essentially empirical in nature, without physical meaning, and therefore are conceptually inferior to the predictive controller. It is a new insight that using closed-loop feedback, prediction is required to physically explain Fitts’ law. The implication is that prediction is an inherent part of the “speed-accuracy trade-off”

Spatial and temporal modulation of cell instructive cues in a filamentous supramolecular biomaterial

Supramolecular materials provide unique opportunities to mimic both the structure and mechanics of the biopolymer networks that compose the extracellular matrix. However, strategies to modify their filamentous structures in space and time in 3D cell culture to study cell behavior as encountered in development and disease are lacking. We herein disclose a multicomponent squaramide-based supramolecular material whose mechanics and bioactivity can be controlled by light through co-assembly of a 1,2-dithiolane (DT) monomer that forms disulfide cross-links. Remarkably, increases in storage modulus from ∼200 Pa to >10 kPa after stepwise photo-cross-linking can be realized without an initiator while retaining colorlessness and clarity. Moreover, viscoelasticity and plasticity of the supramolecular networks decrease upon photo-irradiation, reducing cellular protrusion formation and motility when performed at the onset of cell culture. When applied during 3D cell culture, force-mediated manipulation is impeded and cells move primarily along earlier formed channels in the materials. Additionally, we show photopatterning of peptide cues in 3D using either a photomask or direct laser writing. We demonstrate that these squaramide-based filamentous materials can be applied to the development of synthetic and biomimetic 3D in vitro cell and disease models, where their secondary cross-linking enables mechanical heterogeneity and shaping at multiple length scales

Frequency Locking of an Optical Cavity using LQG Integral Control

This paper considers the application of integral Linear Quadratic Gaussian (LQG) optimal control theory to a problem of cavity locking in quantum optics. The cavity locking problem involves controlling the error between the laser frequency and the resonant frequency of the cavity. A model for the cavity system, which comprises a piezo-electric actuator and an optical cavity is experimentally determined using a subspace identification method. An LQG controller which includes integral action is synthesized to stabilize the frequency of the cavity to the laser frequency and to reject low frequency noise. The controller is successfully implemented in the laboratory using a dSpace DSP board.Comment: 18 pages, 9 figure

Combining Optimal Control Theory and Molecular Dynamics for Protein Folding

A new method to develop low-energy folding routes for proteins is presented. The novel aspect of the proposed approach is the synergistic use of optimal control theory with Molecular Dynamics (MD). In the first step of the method, optimal control theory is employed to compute the force field and the optimal folding trajectory for the atoms of a Coarse-Grained (CG) protein model. The solution of this CG optimization provides an harmonic approximation of the true potential energy surface around the native state. In the next step CG optimization guides the MD simulation by specifying the optimal target positions for the atoms. In turn, MD simulation provides an all-atom conformation whose positions match closely the reference target positions determined by CG optimization. This is accomplished by Targeted Molecular Dynamics (TMD) which uses a bias potential or harmonic restraint in addition to the usual MD potential. Folding is a dynamical process and as such residues make different contacts during the course of folding. Therefore CG optimization has to be reinitialized and repeated over time to accomodate these important changes. At each sampled folding time, the active contacts among the residues are recalculated based on the all-atom conformation obtained from MD. Using the new set of contacts, the CG potential is updated and the CG optimal trajectory for the atoms is recomputed. This is followed by MD. Implementation of this repetitive CG optimization - MD simulation cycle generates the folding trajectory. Simulations on a model protein Villin demonstrate the utility of the method. Since the method is founded on the general tools of optimal control theory and MD without any restrictions, it is widely applicable to other systems. It can be easily implemented with available MD software packages