5,844 research outputs found
Aperiodic colorings and tilings of Coxeter groups
We construct a limit aperiodic coloring of hyperbolic groups. Also we
construct limit strongly aperiodic strictly balanced tilings of the Davis
complex for all Coxeter groups
On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
An improved understanding of the divergence-free constraint for the
incompressible Navier--Stokes equations leads to the observation that a
semi-norm and corresponding equivalence classes of forces are fundamental for
their nonlinear dynamics. The recent concept of {\em pressure-robustness}
allows to distinguish between space discretisations that discretise these
equivalence classes appropriately or not. This contribution compares the
accuracy of pressure-robust and non-pressure-robust space discretisations for
transient high Reynolds number flows, starting from the observation that in
generalised Beltrami flows the nonlinear convection term is balanced by a
strong pressure gradient. Then, pressure-robust methods are shown to outperform
comparable non-pressure-robust space discretisations. Indeed, pressure-robust
methods of formal order are comparably accurate than non-pressure-robust
methods of formal order on coarse meshes. Investigating the material
derivative of incompressible Euler flows, it is conjectured that strong
pressure gradients are typical for non-trivial high Reynolds number flows.
Connections to vortex-dominated flows are established. Thus,
pressure-robustness appears to be a prerequisite for accurate incompressible
flow solvers at high Reynolds numbers. The arguments are supported by numerical
analysis and numerical experiments.Comment: 43 pages, 18 figures, 2 table
Quantitative Assessment of the Anatomical Footprint of the C1 Pedicle Relative to the Lateral Mass: A Guide for C1 Lateral Mass Fixation
Study Design: Anatomic study. Objectives: To determine the relationship of the anatomical footprint of the C1 pedicle relative to the lateral mass (LM). Methods: Anatomic measurements were made on fresh frozen human cadaveric C1 specimens: pedicle width/height, LM width/height (minimum/maximum), LM depth, distance between LMās medial aspect and pedicleās medial border, distance between LMās lateral aspect to pedicleās lateral border, distance between pedicleās inferior aspect and LMās inferior border, distance between archās midline and pedicleās medial border. The percentage of LM medial to the pedicle and the distance from the center of the LM to the pedicleās medial wall were calculated. Results: A total of 42 LM were analyzed. The C1 pedicleās lateral aspect was nearly confluent with the LMās lateral border. Average pedicle width was 9.0 Ā± 1.1 mm, and average pedicle height was 5.0 Ā± 1.1 mm. Average LM width and depth were 17.0 Ā± 1.6 and 17.2 Ā± 1.6 mm, respectively. There was 6.9 Ā± 1.5 mm of bone medial to the medial C1 pedicle, which constituted 41% Ā± 9% of the LMās width. The distance from C1 archās midline to the medial pedicle was 13.5 Ā± 2.0 mm. The LMās center was 1.6 Ā± 1 mm lateral to the medial pedicle wall. There was on average 3.5 Ā± 0.6 mm of the LM inferior to the pedicle inferior border. Conclusions: The center of the lateral mass is 1.6 Ā± 1 mm lateral to the medial wall of the C1 pedicle and approximately 15 mm from the midline. There is 6.9 Ā± 1.5 mm of bone medial to the medial C1 pedicle. Thus, the medial aspect of C1 pedicle may be used as an anatomic reference for locating the center of the C1 LM for screw fixation
Erratum to: Nonpositive Curvature and the Ptolemy Inequality
The following changes would like to be highlighted in the abstract: We provide examples of nonlocally compact, geodesic Ptolemy metric spaces that are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolem
Comparison of Autoscaling Frameworks for Containerised Machine-Learning-Applications in a Local and Cloud Environment
When deploying machine learning (ML) applications, the automated allocation
of computing resources-commonly referred to as autoscaling-is crucial for
maintaining a consistent inference time under fluctuating workloads. The
objective is to maximize the Quality of Service metrics, emphasizing
performance and availability, while minimizing resource costs. In this paper,
we compare scalable deployment techniques across three levels of scaling: at
the application level (TorchServe, RayServe) and the container level (K3s) in a
local environment (production server), as well as at the container and machine
levels in a cloud environment (Amazon Web Services Elastic Container Service
and Elastic Kubernetes Service). The comparison is conducted through the study
of mean and standard deviation of inference time in a multi-client scenario,
along with upscaling response times. Based on this analysis, we propose a
deployment strategy for both local and cloud-based environments.Comment: 6 pages, 3 figure
Multi-scale space-variant FRep cellular structures
Existing mesh and voxel based modeling methods encounter difficulties when dealing with objects containing cellular structures
on several scale levels and varying their parameters in space. We describe an alternative approach based on using real functions evaluated procedurally at any given point. This allows for modeling fully parameterized, nested and multi-scale cellular
structures with dynamic variations in geometric and cellular properties. The geometry of a base unit cell is defined using Function Representation (FRep) based primitives and operations. The unit cell is then replicated in space using periodic
space mappings such as sawtooth and triangle waves. While being replicated, the unit cell can vary its geometry and topology due
to the use of dynamic parameterization. We illustrate this approach by several examples of microstructure generation within a given volume or
along a given surface. We also outline some methods for direct rendering and fabrication not involving auxiliary mesh and voxel
representations
Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier--Stokes equations
Inf-sup stable FEM applied to time-dependent incompressible Navier--Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on an essential regularity assumption for the gradient of the velocity, which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free H1-conforming FEM (like Scott--Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based
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