30 research outputs found
Nonorthogonal Polyhedra Built from Rectangles
We prove that any polyhedron of genus zero or genus one built out of
rectangular faces must be an orthogonal polyhedron, but that there are
nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This
leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].Comment: 19 pages, 20 figures. Revised version makes two corrections: The
statement of the old Lemma 14 was incorrect. It has been corrected and merged
with Lemma 13 now. Second, Figure 19 (a skew quadrilateral) was incorrect,
and is now removed. It played no substantive role in the proof
Binary Space Partitions for Fat Rectangles
This is the published version. Copyright © 2000 Society for Industrial and Applied Mathematic
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Sub-nm wide electron channels protected by topology
Helical locking of spin and momentum and prohibited backscattering are the
key properties of topologically protected states. They are expected to enable
novel types of information processing such as spintronics by providing pure
spin currents, or fault tolerant quantum computation by using the Majorana
fermions at interfaces of topological states with superconductors. So far, the
required helical conduction channels used to realize Majorana fermions are
generated through application of an axial magnetic field to conventional
semiconductor nanowires. Avoiding the magnetic field enhances the possibilities
for circuit design significantly. Here, we show that sub-nanometer wide
electron channels with natural helicity are present at surface step-edges of
the recently discovered topological insulator Bi14Rh3I9. Scanning tunneling
spectroscopy reveals the electron channels to be continuous in both energy and
space within a large band gap of 200 meV, thereby, evidencing its non-trivial
topology. The absence of these channels in the closely related, but
topologically trivial insulator Bi13Pt3I7 corroborates the channels'
topological nature. The backscatter-free electron channels are a direct
consequence of Bi14Rh3I9's structure, a stack of 2D topologically insulating,
graphene-like planes separated by trivial insulators. We demonstrate that the
surface of Bi14Rh3I9 can be engraved using an atomic force microscope, allowing
networks of protected channels to be patterned with nm precision.Comment: 17 pages, 4 figures, and supplementary material, Nature Physics in
pres
(Non)Existence of Pleated Folds: How Paper Folds Between Creases
We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the model can be folded with additional creases, suggesting that real paper “folds” into this model via small such creases. We conjecture that the circular version of this model, consisting simply of concentric circular creases, also folds without extra creases. At the heart of our results is a new structural theorem characterizing uncreased intrinsically flat surfaces—the portions of paper between the creases. Differential geometry has much to say about the local behavior of such surfaces when they are sufficiently smooth, e.g., that they are torsal ruled. But this classic result is simply false in the context of the whole surface. Our structural characterization tells the whole story, and even applies to surfaces with discontinuities in the second derivative. We use our theorem to prove fundamental properties about how paper folds, for example, that straight creases on the piece of paper must remain piecewise-straight (polygonal) by folding.National Science Foundation (U.S.) (CAREER Award CCF-0347776
Online Data Structures in External Memory
The original publication is available at www.springerlink.comThe data sets for many of today's computer applications are
too large to t within the computer's internal memory and must instead
be stored on external storage devices such as disks. A major performance
bottleneck can be the input/output communication (or I/O) between
the external and internal memories. In this paper we discuss a variety of
online data structures for external memory, some very old and some very
new, such as hashing (for dictionaries), B-trees (for dictionaries and 1-D
range search), bu er trees (for batched dynamic problems), interval trees
with weight-balanced B-trees (for stabbing queries), priority search trees
(for 3-sided 2-D range search), and R-trees and other spatial structures.
We also discuss several open problems along the way
A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes
Copyright © 2015 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics Vol. 290 (2015), DOI: 10.1016/j.jcp.2015.02.045A new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.Natural Environment Research Council under the “GungHo” projec
Optimal System Design with Geometric Considerations.
System design is tied to both functionality and geometric realization. The former is pertinent to system performance, and the latter is related to packaging. Packaging is an optimization process that finds a desirable placement for the system components within a given space. When the components do not fit into the allocated space at the packaging stage, the design engineers must make modifications that can affect the performance of the system. The modification of a component can also affect the geometry and positions of other components in the system. These changes might lead to an infeasible layout. Therefore, optimizing the system performance considering packaging is desirable.
Packaging problems and solution methods have been studied in many applications, such as electrical circuit layout, glass or metal cutting, truck loading, trunk packing, rapid prototyping (RP), architectural floor plan layout, routing, and mechanical component layout. Packaging problems in a mechanical system design are more challenging than 2D applications such as circuit layout and the metal cutting problem; this is due to a larger design space and increased complexity of geometry. Complex 3D geometry leads to increased computational time for interference checking, which is inevitable for finding a feasible layout. Detailed 3D CAD models, however, are not required or not available at the preliminary design stage. Therefore, abstract representation of the components is necessary during the layout process. Abstract models should balance accuracy of geometry representation and rapid computation capturing designers’ intent.
This dissertation presents a computational environment for addressing the combined packaging and optimal system design. The packaging problem also includes pipe generation because pipe routing is also important problems in mechanical system design. The simulation model of a thermal management system for heavy duty series hybrid electric vehicles is used to demonstrate the usefulness of the proposed framework.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108865/1/kwangjae_1.pd