6,347 research outputs found
Comment on "Jamming at zero temperature and zero applied stress: The epitome of disorder"
O'Hern, Silbert, Liu and Nagel [Phys. Rev. E. 68, 011306 (2003)] (OSLN) claim
that a special point of a "jamming phase diagram" (in density, temperature,
stress space) is related to random close packing of hard spheres, and that it
represents, for their suggested definitions of jammed and random, the recently
introduced maximally random jammed state. We point out several difficulties
with their definitions and question some of their claims. Furthermore, we
discuss the connections between their algorithm and other hard-sphere packing
algorithms in the literature.Comment: 4 pages of text, already publishe
Epitaxial Frustration in Deposited Packings of Rigid Disks and Spheres
We use numerical simulation to investigate and analyze the way that rigid
disks and spheres arrange themselves when compressed next to incommensurate
substrates. For disks, a movable set is pressed into a jammed state against an
ordered fixed line of larger disks, where the diameter ratio of movable to
fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations
is 0.7, where the fixed substrate has the structure of a (001) plane of a
face-centered cubic array. Results obtained for both disks and spheres exhibit
various forms of density-reducing packing frustration next to the
incommensurate substrate, including some cases displaying disorder that extends
far from the substrate. The disk system calculations strongly suggest that the
most efficient (highest density) packings involve configurations that are
periodic in the lateral direction parallel to the substrate, with substantial
geometric disruption only occurring near the substrate. Some evidence has also
emerged suggesting that for the sphere systems a corresponding structure doubly
periodic in the lateral directions would yield the highest packing density;
however all of the sphere simulations completed thus far produced some residual
"bulk" disorder not obviously resulting from substrate mismatch. In view of the
fact that the cases studied here represent only a small subset of all that
eventually deserve attention, we end with discussion of the directions in which
first extensions of the present simulations might profitably be pursued.Comment: 28 pages, 14 figures; typos fixed; a sentence added to 4th paragraph
of sect 5 in responce to a referee's comment
Hypoconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids
Continuing on recent computational and experimental work on jammed packings
of hard ellipsoids [Donev et al., Science, vol. 303, 990-993] we consider
jamming in packings of smooth strictly convex nonspherical hard particles. We
explain why the isocounting conjecture, which states that for large disordered
jammed packings the average contact number per particle is twice the number of
degrees of freedom per particle (\bar{Z}=2d_{f}), does not apply to
nonspherical particles. We develop first- and second-order conditions for
jamming, and demonstrate that packings of nonspherical particles can be jammed
even though they are hypoconstrained (\bar{Z}<2d_{f}). We apply an algorithm
using these conditions to computer-generated hypoconstrained ellipsoid and
ellipse packings and demonstrate that our algorithm does produce jammed
packings, even close to the sphere point. We also consider packings that are
nearly jammed and draw connections to packings of deformable (but stiff)
particles. Finally, we consider the jamming conditions for nearly spherical
particles and explain quantitatively the behavior we observe in the vicinity of
the sphere point.Comment: 33 pages, third revisio
Locked and Unlocked Chains of Planar Shapes
We extend linkage unfolding results from the well-studied case of polygonal
linkages to the more general case of linkages of polygons. More precisely, we
consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are
hinged together sequentially at rotatable joints. Our goal is to characterize
the families of planar shapes that admit locked chains, where some
configurations cannot be reached by continuous reconfiguration without
self-intersection, and which families of planar shapes guarantee universal
foldability, where every chain is guaranteed to have a connected configuration
space. Previously, only obtuse triangles were known to admit locked shapes, and
only line segments were known to guarantee universal foldability. We show that
a surprisingly general family of planar shapes, called slender adornments,
guarantees universal foldability: roughly, the distance from each edge along
the path along the boundary of the slender adornment to each hinge should be
monotone. In contrast, we show that isosceles triangles with any desired apex
angle less than 90 degrees admit locked chains, which is precisely the
threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof
details. (Fixed crash-induced bugs in the abstract.
Algebra versus analysis in the theory of flexible polyhedra
Two basic theorems of the theory of flexible polyhedra were proven by
completely different methods: R. Alexander used analysis, namely, the Stokes
theorem, to prove that the total mean curvature remains constant during the
flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to
prove that the oriented volume remains constant during the flex. We show that
none of these methods can be used to prove the both theorems. As a by-product,
we prove that the total mean curvature of any polyhedron in the Euclidean
3-space is not an algebraic function of its edge lengths.Comment: 5 pages, 5 figures; condition (iii) in Theorem 5 is correcte
When do we eat? An evaluation of food items input into an electronic monitoring application
We present a formative study that examines what, when, and how participants in a chronic kidney disease (stage 5) population input food items into an electronic intake monitoring application. Participants scanned food item barcodes or voice recorded food items they consumed during a three week period. The results indicated that a learning curve was associated with barcode scanning; participants with low literacy skills had difficulty describing food items in voice recordings; and participants input food items depending on when they had dialysis treatment. Participants thought this electronic self monitoring application would be helpful for chronically ill populations in their first year of treatmen
The orbit rigidity matrix of a symmetric framework
A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks. The orbit matrix is a true
analog of the standard rigidity matrix for general frameworks, and its analysis
gives important insights into questions about the flexibility and rigidity of
classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the
power to predict symmetry-preserving finite mechanisms - giving a simplified
analysis which covers a wide range of the known mechanisms, and generalizes the
classes of known mechanisms. This initial exploration of the properties of the
orbit matrix also opens up a number of new questions and possible extensions of
the previous results, including transfer of symmetry based results from
Euclidean space to spherical, hyperbolic, and some other metrics with shared
symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure
Forms of Knowledge of Advanced Mathematics for Teaching
In this paper, we explore in more detail why knowing advanced mathematics might be beneficial for teachers, specifically in relation to their classroom practice. Rather than by listing courses or specific advanced topics, as though those were the agents of change, we do so by considering advanced mathematical content for teachers in terms of more general forms of knowledge. In particular, we identify five forms of knowledge of advanced mathematics for teaching: peripheral, evolutionary, axiomatic, logical, and inferential. These categories were derived from analysis of an extensive mapping process linking K-12 content to relevant advanced mathematics. We connect these forms of knowledge to particular practices that teachers engage in so as to clarify the perceived relations to classroom practices. We view such work as important to and productive for teacher educators, particularly in conceptualizing and structuring mathematics courses for teachers so that content that truly informs the work of K-12 teaching can be highlighted, and in a manner that facilitates teachers’ formation of those connections
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