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A fibred hyperbolic 3-manifold induces a map from the hyperbolic plane to hyperbolic 3-space, the respective universal covers of the fibre, and the manifold. The induced map is an embedding that is exponentially distorted in terms of the individual metrics. In this article, we begin a study of the distortion along typical rays in the fibre. We verify that a typical ray in the hyperbolic plane makes linear progress in the ambient metric in hyperbolic 3-space. We formulate the proof in terms of some soft aspects of the geometry and basic ergodic theory. This enables us to extend the result to analogous contexts that correspond to certain extensions of closed surface groups. These include surface group extensions that are Gromov hyperbolic, the universal curve over a Teichmüller disc, and the extension induced by the Birman exact sequence. [ABSTRACT FROM AUTHOR]

We investigate the sharp interface limit of a diffuse interface system that couples the Allen--Cahn equation with the instationary Navier--Stokes system in a bounded domain in Rd with d Є{ 2, 3}. This model is used to describe a propagating front in a viscous incompressible flow with the width of the transition layer being characterized by a small parameter ε > 0. We show that the solutions converge to a limit two-phase fluid system with surface tension that couples the mean curvature flow and the Navier--Stokes system. The main assumptions are that the evolution of the limit system is sufficiently regular and that the associated evolving interface does not intersect the boundary of the container. For quantitatively well-prepared initial data, we even establish an optimal convergence rate. This is the first rigorous result of this kind which is valid in all physically relevant ambient dimensions. [ABSTRACT FROM AUTHOR]

Let S be a closed, genus g surface. The space of geodesic currents on S encompasses the set of closed curves up to homotopy, as well as Teichmüller space, and many other spaces of structures on S. We show that one can define a mapping class group equivariant, length minimizing projection from the set of filling geodesic currents down to Teichmüller space, and prove some basic properties of this projection to show that it is well-behaved. [ABSTRACT FROM AUTHOR]

We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient flow calibration in the sense of the recent work of Fischer et al. (2020) for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions. [ABSTRACT FROM AUTHOR]

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago–Ivanov–Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group. [ABSTRACT FROM AUTHOR]

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus. [ABSTRACT FROM AUTHOR]

Many living beings show the ability and necessity to develop invertible, tubular structures to enable additional functions temporarily. The biological archetypes always demonstrate a high change of volume of the structure between an inactive and active state. This makes the principle interesting for many technical applications, where a certain geometry or an additional volume has to be generated situationally for a task and can only be accepted temporarily, for example, in minimally invasive robotics. A possibility was sought to transfer the archetype into the technical context and to evaluate geometric-constructive dependencies based on an inversion of the structure. The result is a practicable design for repeatedly invertible net structures, which can be used for products with temporary additional functions and volumes. [ABSTRACT FROM AUTHOR]

In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities, the concept of varifold solutions—whose global in time existence has been shown by Abels (Interfaces Free Bound 9(1):31–65, 2007) for general initial data—does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy. [ABSTRACT FROM AUTHOR]

The potential of adaptronic applications has been proven in many conceptual studies. A broad use in high-efficiency branches is often hindered by the absence of an appropriate assembly method. Especially for piezoceramic foil transducers, the application on structural parts can be simplified using a semi-finished part that includes the transducer. The part is then shaped in a final forming operation. The purpose of the present study is the investigation of process limits in automated process chains for producing semi-finished parts. An adhesive is used in the process, which is only locally cured. This bi-conditioned state is achieved using cooling and heating elements. The process limits are mainly affected by the choice of temperature and curing time between adhesive application and forming operation. Several tests with a rotational rheometer were carried out to investigate the curing behavior. An appropriate process window was identified varying processing time and temperature. The results were then used to build a model of the curing behavior. A mathematical approach had to be used to find the best configuration because no sharp border exists between the two adhesive conditions of liquid and solid state. The process parameters were proven with runs inside and outside of the process limits. [ABSTRACT FROM AUTHOR]

Let $\unicode[STIX]{x1D70B}$ be a non-degenerate permutation on at least four symbols. We show that the set of uniquely ergodic interval exchange transformations with permutation $\unicode[STIX]{x1D70B}$ is path-connected. [ABSTRACT FROM AUTHOR]

We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph Γ, and any finite group G⩽U0(AΓ)⩽ Out 0(AΓ), we find a non‐positively curved cube complex with fundamental group AΓ on which G acts by isometries, realising the action on AΓ. [ABSTRACT FROM AUTHOR]

We show that a random |$3$| -manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth. [ABSTRACT FROM AUTHOR]

The technique joining by forming allows the structural integration of piezoceramic fibers into locally microstructured metal sheets without any elastic interlayers. A high-volume production of the joining partners causes in statistical deviations from the nominal dimensions. A numerical simulation on geometric process sensitivity shows that the deviations have a high significant influence on the resulting fiber stresses after the joining by forming operation and demonstrate the necessity of a monitoring concept. On this basis, the electromechanical behavior of piezoceramic array transducers is investigated experimentally before, during and after the joining process. The piezoceramic array transducer consists of an arrangement of five electrical interconnected piezoceramic fibers. The findings show that the impedance spectrum depends on the fiber stresses and can be used for in-process monitoring during the joining process. Based on the impedance values the preload state of the interconnected piezoceramic fibers can be specifically controlled and a fiber overload. [ABSTRACT FROM AUTHOR]

Consider a finite, regular cover Y → X of finite graphs, with associated deck group G. We relate the topology of the cover to the structure of H1(Y;ℂ) as a G-representation. A central object in this study is the primitive homology group Hprim1 (Y;ℂ) ⊆ H1(Y ;ℂ), which is the span of homology classes represented by components of lifts of primitive elements of π1(X). This circle of ideas relates combinatorial group theory, surface topology, and representation theory. [ABSTRACT FROM AUTHOR]

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces. [ABSTRACT FROM AUTHOR]

The outer automorphism group $\mathrm {Out}(F_{2g})$ of a free group on $2g$ generators naturally contains the mapping class group of a punctured genus $g$ surface $S_{g,1}$ as a subgroup. We define a ‘subsurface projection’ of the sphere complex of the connected sum of $n$ copies of $S^1\times S^2$ into the arc complex of $S_{g,1}$. Using this, we show that $\mathrm {Map} (S_{g,1})$ is a Lipschitz retract of $\mathrm {Out}(F_{2g})$. We use another ‘subsurface projection’ to give a simple proof of a result of Handel and Mosher [‘Lipschitz retraction and distortion for subgroups of $\mathrm {Out}(F_n)$’, Geom. Topol. 17 (2013) 1535–1580] stating that stabilizers of conjugacy classes of free splittings and corank $1$ free factors in a free group $F_n$ are Lipschitz retracts of $\mathrm {Out}(F_n)$. [ABSTRACT FROM PUBLISHER]

Abstract: With the objective of producing aluminum sandwich parts with a sensor and actuator functionality, a production method was proposed which uses an adhesive for the integration of a piezomodule. Complex shapes may cause a reduction in the performance of piezomodules. Uniaxial and biaxial transducer tensile tests and accompanying optical strain measurements were carried out to evaluate the residual performance taking into account the strain load combination. The model validation showed a very good agreement between forming experiment and numerical simulation. The simulation model including the functionality reduction enables the performance of the actuator to be predicted after the forming stage for complex shaped parts has been completed. [Copyright &y& Elsevier]

We prove that the ending lamination space of the five-punctured sphere is homeomorphic to the Nöbeling curve. [ABSTRACT FROM AUTHOR]

The integration of sensors and actuators in material‐compounds allows for additional functionalities within minimum space. Especially, piezoelectric materials are widely used in industrial applications, such as fuel injection or positioning. Up to now, production of sheet‐metal‐compounds with a sensor and actuator functionality is strongly restricted by manual fabrication processes. The design of a material‐compound including aluminum sheet metals and flat piezoceramic fibers (piezoceramic‐metal‐compound) and a process chain allowing for an automated production offers the possibility of serial production. The compound design, the process chain for the fabrication of semi‐finished parts, the forming into structural parts, and the functionality are presented. The process chain enables piezoceramic‐metal‐compounds for large‐scale production. Piezoceramic‐metal‐compounds are used as sensors and actuators. The authors propose a forming process chain that provides semi‐finished parts with integrated piezomodules to shorten cycle times. Furthermore, an adhesive is utilized that partly remains in a liquid condition when forming takes place. A proper temperature–time regime is identified. Finally, the forming limits of the process are investigated. [ABSTRACT FROM AUTHOR]

The gear whine phenomenon in vehicles is a well‐known acoustic issue which often needs a development of individual measures. Especially in rear‐wheel and four‐wheel drive vehicles, the acoustical optimization of the rear axle leads to the major goal conflict: improvement of the vibrational isolation of the rear axle versus excellent driving dynamics of the vehicle. A promising solution for this issue is described in this paper. The key objective is to reduce the noise contributions of the rear axle using an active vibration control system (AVC) based on piezoelectric actuators on the transfer path from the axle into the vehicle. In the first part of the paper, the development process of the AVC system is described. In the second part of the paper, a rear axle test bench is presented which is used to validate the simulation results. A multi component control algorithm has been implemented in order to get an optimum reduction of relevant forces at the mounting points to the vehicle using one or more actuators. In preparation for a series launch of the AVC system, an assembly concept for applying the piezoelectric actuators to the rear sub frame in a series production process is presented. This article presents the development of an active vibration control system for compensation of disturbing gear whine noises of a rear axle using piezoelectric actuators. The system is applied on the rear axle's sub frame in order to influence structure‐borne noise transfer in the mechanical structure which is caused by the excitation of the gear mesh. [ABSTRACT FROM AUTHOR]