3,651 research outputs found
(2+1) gravity for higher genus in the polygon model
We construct explicitly a (12g-12)-dimensional space P of unconstrained and
independent initial data for 't Hooft's polygon model of (2+1) gravity for
vacuum spacetimes with compact genus-g spacelike slices, for any g >= 2. Our
method relies on interpreting the boost parameters of the gluing data between
flat Minkowskian patches as the lengths of certain geodesic curves of an
associated smooth Riemann surface of the same genus. The appearance of an
initial big-bang or a final big-crunch singularity (but never both) is verified
for all configurations. Points in P correspond to spacetimes which admit a
one-polygon tessellation, and we conjecture that P is already the complete
physical phase space of the polygon model. Our results open the way for
numerical investigations of pure (2+1) gravity.Comment: 35 pages, 22 figure
Meeting in a Polygon by Anonymous Oblivious Robots
The Meeting problem for searchers in a polygon (possibly with
holes) consists in making the searchers move within , according to a
distributed algorithm, in such a way that at least two of them eventually come
to see each other, regardless of their initial positions. The polygon is
initially unknown to the searchers, and its edges obstruct both movement and
vision. Depending on the shape of , we minimize the number of searchers
for which the Meeting problem is solvable. Specifically, if has a
rotational symmetry of order (where corresponds to no
rotational symmetry), we prove that searchers are sufficient, and
the bound is tight. Furthermore, we give an improved algorithm that optimally
solves the Meeting problem with searchers in all polygons whose
barycenter is not in a hole (which includes the polygons with no holes). Our
algorithms can be implemented in a variety of standard models of mobile robots
operating in Look-Compute-Move cycles. For instance, if the searchers have
memory but are anonymous, asynchronous, and have no agreement on a coordinate
system or a notion of clockwise direction, then our algorithms work even if the
initial memory contents of the searchers are arbitrary and possibly misleading.
Moreover, oblivious searchers can execute our algorithms as well, encoding
information by carefully positioning themselves within the polygon. This code
is computable with basic arithmetic operations, and each searcher can
geometrically construct its own destination point at each cycle using only a
compass. We stress that such memoryless searchers may be located anywhere in
the polygon when the execution begins, and hence the information they initially
encode is arbitrary. Our algorithms use a self-stabilizing map construction
subroutine which is of independent interest.Comment: 37 pages, 9 figure
Wonder of sine-Gordon Y-systems
The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were
introduced by Tateo in the 90's in the study of the integrable deformation of
conformal field theory by the thermodynamic Bethe ansatz method. The
periodicity property and the dilogarithm identities concerning these Y-systems
were conjectured by Tateo, and only a part of them have been proved so far. In
this paper we formulate these Y-systems by the polygon realization of cluster
algebras of types A and D, and prove the conjectured periodicity and
dilogarithm identities in full generality. As it turns out, there is a
wonderful interplay among continued fractions, triangulations of polygons,
cluster algebras, and Y-systems.Comment: v1: 66 pages; v2: 53 pages, the version to appear in Trans. Amer.
Math. Soc. (in the journal version, the proofs of Props. 5.29-5.31 and Sect.
5.8 will be omitted due to the limitation of space); v3: 53 pages, minor
improvement of figures; v4 (no text changes): Sage (v7.0 and higher) has
built-in functions to plot the triangulations associated with sine-Gordon and
reduced sine-Gordon Y-system
Triangles bridge the scales: Quantifying cellular contributions to tissue deformation
In this article, we propose a general framework to study the dynamics and
topology of cellular networks that capture the geometry of cell packings in
two-dimensional tissues. Such epithelia undergo large-scale deformation during
morphogenesis of a multicellular organism. Large-scale deformations emerge from
many individual cellular events such as cell shape changes, cell
rearrangements, cell divisions, and cell extrusions. Using a triangle-based
representation of cellular network geometry, we obtain an exact decomposition
of large-scale material deformation. Interestingly, our approach reveals
contributions of correlations between cellular rotations and elongation as well
as cellular growth and elongation to tissue deformation. Using this Triangle
Method, we discuss tissue remodeling in the developing pupal wing of the fly
Drosophila melanogaster.Comment: 26 pages, 18 figure
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Subquadratic nonobtuse triangulation of convex polygons
A convex polygon with n sides can be triangulated by O(n^1.85) triangles, without any obtuse angles. The construction uses a novel form of geometric divide and conquer
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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
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