7,274 research outputs found
Local conditions for exponentially many subdivisions
Given a graph F, let st(F) be the number of subdivisions of F, each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F. In 1990, Tuza asked for which graphs F and large t, one has that st(F) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st(F) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterize those F for which st(F) is exponential in a power of t
Isotopic Equivalence from Bezier Curve Subdivision
We prove that the control polygon of a Bezier curve B becomes homeomorphic
and ambient isotopic to B via subdivision, and we provide closed-form formulas
to compute the number of iterations to ensure these topological
characteristics. We first show that the exterior angles of control polygons
converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035
How Close to Two Dimensions Does a Lennard-Jones System Need to Be to Produce a Hexatic Phase?
We report on a computer simulation study of a Lennard-Jones liquid confined
in a narrow slit pore with tunable attractive walls. In order to investigate
how freezing in this system occurs, we perform an analysis using different
order parameters. Although some of the parameters indicate that the system goes
through a hexatic phase, other parameters do not. This shows that to be certain
whether a system has a hexatic phase, one needs to study not only a large
system, but also several order parameters to check all necessary properties. We
find that the Binder cumulant is the most reliable one to prove the existence
of a hexatic phase. We observe an intermediate hexatic phase only in a
monolayer of particles confined such that the fluctuations in the positions
perpendicular to the walls are less then 0.15 particle diameters, i. e. if the
system is practically perfectly 2d
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Comparison of Current Gravity Estimation and Determination Models
This paper will discuss the history of gravity estimation and determination models while analyzing methods that are in development. Some fundamental methods for calculating the gravity field include spherical harmonics solutions, local weighted interpolation, and global point mascon modeling (PMC). Recently, high accuracy measurements have become more accessible, and the requirements for high order geopotential modeling have become more stringent. Interest in irregular bodies, accurate models of the hydrological system, and on-board processing has demanded a comprehensive model that can quickly and accurately compute the geopotential with low memory costs. This trade study of current geopotential modeling techniques will reveal that each modeling technique has a unique use case. It is notable that the spherical harmonics model is relatively accurate but poses a cumbersome inversion problem. PMC and interpolation models, on the other hand, are computationally efficient, but require more research to become robust models with high levels of accuracy. Considerations of the trade study will suggest further research for the point mascon model. The PMC model should be improved through mascon refinement, direct solutions that stem from geodetic measurements, and further validation of the gravity gradient. Finally, the potential for each model to be implemented with parallel computation will be shown to lead to large improvements in computing time while reducing the memory cost for each technique.Aerospace Engineering and Engineering Mechanic
Tubular Neighborhoods of Nodal Sets and Diophantine Approximation
We give upper and lower bounds on the volume of a tubular neighborhood of the
nodal set of an eigenfunction of the Laplacian on a real analytic closed
Riemannian manifold M. As an application we consider the question of
approximating points on M by nodal sets, and explore analogy with approximation
by rational numbers.Comment: 22 pages; revised version containing full proof of lower bound;
reference added; to appear in Amer. J. Math
Digraphs and cycle polynomials for free-by-cyclic groups
Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be
represented by an expanding, irreducible train-track map. The automorphism
determines a free-by-cyclic group
and a homomorphism . By work of Neumann,
Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, has an open cone
neighborhood in whose integral points
correspond to other fibrations of whose associated outer automorphisms
are themselves representable by expanding irreducible train-track maps. In this
paper, we define an analog of McMullen's Teichm\"uller polynomial that computes
the dilatations of all outer automorphism in .Comment: 41 pages, 20 figure
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