788 research outputs found
Tiling by rectangles and alternating current
This paper is on tilings of polygons by rectangles. A celebrated physical
interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and
W.T. Tutte uses direct-current circuits. The new approach of the paper is an
application of alternating-current circuits.
The following results are obtained:
- a necessary condition for a rectangle to be tilable by rectangles of given
shapes;
- a criterion for a rectangle to be tilable by rectangles similar to it but
not all homothetic to it;
- a criterion for a generic polygon to be tilable by squares.
These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich,
D. Rinne and G. Szekeres.Comment: In English and in Russian; 21 pages; 6 figures; minor improvement of
exposition, Russian translation adde
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
IMITATOR II: A Tool for Solving the Good Parameters Problem in Timed Automata
We present here Imitator II, a new version of Imitator, a tool implementing
the "inverse method" for parametric timed automata: given a reference valuation
of the parameters, it synthesizes a constraint such that, for any valuation
satisfying this constraint, the system behaves the same as under the reference
valuation in terms of traces, i.e., alternating sequences of locations and
actions. Imitator II also implements the "behavioral cartography algorithm",
allowing us to solve the following good parameters problem: find a set of
valuations within a given bounded parametric domain for which the system
behaves well. We present new features and optimizations of the tool, and give
results of applications to various examples of asynchronous circuits and
communication protocols.Comment: In Proceedings INFINITY 2010, arXiv:1010.611
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Generation of Porous Structures Using Fused Deposition
The Fused Deposition Modeling process uses hardware and software machine-level
language that are very similar to that of a pen-plotter. Consequently, the·use of patterns with
poly-lines as basic geometric features, instead of the current method based on filled polygons
(monolithic models), can increase its efficiency.
In the current study, various toolpath planning methods have been developed to fabricate
porous structures. Computational domain decomposition methods can be applied to the physical
or to slice-level domains to generate structured and unstructured grids. Also, textures can be
created using periodic tiling of the layer with unit cells (squares, honeycombs, etc). Methods
'based on curves include fractal space filling curves and.change of effective road width Within a
layer or within a continuous curve. Individual phases can also be placed in binary compositions.
In present investigation, a custom software has been developed and implemented to
generate build files (SML) and slice files (SSL) for the above-mentioned structures, demonstrating the efficient control ofthe size, shape, and distribution ofporosity.Mechanical Engineerin
Synthetic magnetic fluxes on the honeycomb lattice
We devise experimental schemes able to mimic uniform and staggered magnetic
fluxes acting on ultracold two-electron atoms, such as ytterbium atoms,
propagating in a honeycomb lattice. The atoms are first trapped into two
independent state-selective triangular lattices and are further exposed to a
suitable configuration of resonant Raman laser beams. These beams induce hops
between the two triangular lattices and make atoms move in a honeycomb lattice.
Atoms traveling around each unit cell of this honeycomb lattice pick up a
nonzero phase. In the uniform case, the artificial magnetic flux sustained by
each cell can reach about two flux quanta, thereby realizing a cold atom
analogue of the Harper model with its notorious Hofstadter's butterfly
structure. Different condensed-matter phenomena such as the relativistic
integer and fractional quantum Hall effects, as observed in graphene samples,
could be targeted with this scheme.Comment: 12 pages, 14 figure
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