3,413 research outputs found
Curves of Finite Total Curvature
We consider the class of curves of finite total curvature, as introduced by
Milnor. This is a natural class for variational problems and geometric knot
theory, and since it includes both smooth and polygonal curves, its study shows
us connections between discrete and differential geometry. To explore these
ideas, we consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.Comment: 25 pages, 4 figures; final version, to appear in "Discrete
Differential Geometry", Oberwolfach Seminars 38, Birkhauser, 200
Calculation of the incremental stress-strain relation of a polygonal packing
The constitutive relation of the quasi-static deformation on two dimensional
packed samples of polygons is calculated using molecular dynamic simulations.
The stress values at which the system remains stable are bounded by a failure
surface, that shows a power law dependence on the pressure. Below the failure
surface, non linear elasticity and plastic deformation are obtained, which are
evaluated in the framework of the incremental linear theory. The results shows
that the stiffness tensor can be directly related to the micro-contact
rearrangements. The plasticity obeys a non-associated flow rule, with a plastic
limit surface that does not agree with the failure surface.Comment: 11 pages, 20 figur
DNA folding: structural and mechanical properties of the two-angle model for chromatin
We present a theoretical analysis of the structural and mechanical properties
of the 30-nm chromatin fiber. Our study is based on the two-angle model
introduced by Woodcock et al. (Woodcock, C. L., S. A. Grigoryev, R. A.
Horowitz, and N. Whitaker. 1993. PNAS 90:9021-9025) that describes the
chromatin fiber geometry in terms of the entry-exit angle of the nucleosomal
DNA and the rotational setting of the neighboring nucleosomes with respect to
each other. We explore analytically the different structures that arise from
this building principle, and demonstrate that the geometry with the highest
density is close to the one found in native chromatin fibers under
physiological conditions. On the basis of this model we calculate mechanical
properties of the fiber under stretching. We obtain expressions for the
stress-strain characteristics which show good agreement with the results of
recent stretching experiments (Cui, Y., and C. Bustamante. 2000. PNAS
97:127-132) and computer simulations (Katritch, V., C. Bustamante, and W. K.
Olson. 2000. J. Mol. Biol. 295:29-40), and which provide simple physical
insights into correlations between the structural and elastic properties of
chromatin.Comment: 23 pages, 6 figures, to be published in Biophys.
Accelerating Reinforcement Learning by Composing Solutions of Automatically Identified Subtasks
This paper discusses a system that accelerates reinforcement learning by
using transfer from related tasks. Without such transfer, even if two tasks are
very similar at some abstract level, an extensive re-learning effort is
required. The system achieves much of its power by transferring parts of
previously learned solutions rather than a single complete solution. The system
exploits strong features in the multi-dimensional function produced by
reinforcement learning in solving a particular task. These features are stable
and easy to recognize early in the learning process. They generate a
partitioning of the state space and thus the function. The partition is
represented as a graph. This is used to index and compose functions stored in a
case base to form a close approximation to the solution of the new task.
Experiments demonstrate that function composition often produces more than an
order of magnitude increase in learning rate compared to a basic reinforcement
learning algorithm
Fragmentation of a Circular Disc by Impact on a Frictionless Plate
The break-up of a two-dimensional circular disc by normal and oblique impact
on a hard frictionless plate is investigated by molecular dynamics simulations.
The disc is composed of numerous unbreakable randomly shaped convex polygons
connected together by simple elastic beams that break when bent or stretched
beyond a certain limit. It is found that for both normal and oblique impacts
the crack patterns are the same and depend solely on the normal component of
the impact velocity. Analysing the pattern of breakage, amount of damage,
fragment masses and velocities, we show the existence of a critical velocity
which separates two regimes of the impact process: below the critical point
only a damage cone is formed at the impact site (damage), cleaving of the
particle occurs at the critical point, while above the critical velocity the
disc breaks into several pieces (fragmentation). In the limit of very high
impact velocities the disc suffers complete disintegration (shattering) into
many small fragments. In agreement with experimental results, fragment masses
are found to follow the Gates-Gaudin-Schuhmann distribution (power law) with an
exponent independent of the velocity and angle of impact. The velocity
distribution of fragments exhibit an interesting anomalous scaling behavior
when changing the impact velocity and the size of the disc.Comment: submitted to J. Phys: Condensed Matter special issue on Granular
Medi
Implementing vertex dynamics models of cell populations in biology within a consistent computational framework
The dynamic behaviour of epithelial cell sheets plays a central role during development, growth, disease and wound healing. These processes occur as a result of cell adhesion, migration, division, differentiation and death, and involve multiple processes acting at the cellular and molecular level. Computational models offer a useful means by which to investigate and test hypotheses about these processes, and have played a key role in the study of cellâcell interactions. However, the necessarily complex nature of such models means that it is difficult to make accurate comparison between different models, since it is often impossible to distinguish between differences in behaviour that are due to the underlying model assumptions, and those due to differences in the in silico implementation of the model. In this work, an approach is described for the implementation of vertex dynamics models, a discrete approach that represents each cell by a polygon (or polyhedron) whose vertices may move in response to forces. The implementation is undertaken in a consistent manner within a single open source computational framework, Chaste, which comprises fully tested, industrial-grade software that has been developed using an agile approach. This framework allows one to easily change assumptions regarding force generation and cell rearrangement processes within these models. The versatility and generality of this framework is illustrated using a number of biological examples. In each case we provide full details of all technical aspects of our model implementations, and in some cases provide extensions to make the models more generally applicable
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Stenomaps: Shorthand for shapes
We address some of the challenges in representing spatial data with a novel form of geometric abstraction-the stenomap. The stenomap comprises a series of smoothly curving linear glyphs that each represent both the boundary and the area of a polygon. We present an efficient algorithm to automatically generate these open, C1-continuous splines from a set of input polygons. Feature points of the input polygons are detected using the medial axis to maintain important shape properties. We use dynamic programming to compute a planar non-intersecting spline representing each polygon's base shape. The results are stylised glyphs whose appearance may be parameterised and that offer new possibilities in the 'cartographic design space'. We compare our glyphs with existing forms of geometric schematisation and discuss their relative merits and shortcomings. We describe several use cases including the depiction of uncertain model data in the form of hurricane track forecasting; minimal ink thematic mapping; and the depiction of continuous statistical data
A Survey of Developable Surfaces: From Shape Modeling to Manufacturing
Developable surfaces are commonly observed in various applications such as
architecture, product design, manufacturing, and mechanical materials, as well
as in the development of tangible interaction and deformable robots, with the
characteristics of easy-to-product, low-cost, transport-friendly, and
deformable. Transforming shapes into developable surfaces is a complex and
comprehensive task, which forms a variety of methods of segmentation,
unfolding, and manufacturing for shapes with different geometry and topology,
resulting in the complexity of developable surfaces. In this paper, we reviewed
relevant methods and techniques for the study of developable surfaces,
characterize them with our proposed pipeline, and categorize them based on
digital modeling, physical modeling, interaction, and application. Through the
analysis to the relevant literature, we also discussed some of the research
challenges and future research opportunities.Comment: 20 pages, 24 figures, Author submitted manuscrip
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