312 research outputs found
Diameter graphs of polygons and the proof of a conjecture of Graham
AbstractWe show that for an n-gon with unit diameter to have maximum area, its diameter graph must contain a cycle, and we derive an isodiametric theorem for such n-gons in terms of the length of the cycle. We then apply this theorem to prove Graham's 1975 conjecture that the diameter graph of a maximal 2m-gon (mâ©Ÿ3) must be a cycle of length 2mâ1 with one additional edge attached to it
Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in
small polygons which have the largest area for a given number of vertices .
Many instances are already solved in the literature, namely for all odd ,
and for and 8. Thus, for even , instances of this problem
remain open. Finding those largest small polygons can be formulated as
nonconvex quadratic programming problems which can challenge state-of-the-art
global optimization algorithms. We show that a recently developed technique for
global polynomial optimization, based on a semidefinite programming approach to
the generalized problem of moments and implemented in the public-domain Matlab
package GloptiPoly, can successfully find largest small polygons for and
. Therefore this significantly improves existing results in the domain.
When coupled with accurate convex conic solvers, GloptiPoly can provide
numerical guarantees of global optimality, as well as rigorous guarantees
relying on interval arithmetic
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons
A small polygon is a polygon of unit diameter. The maximal area of a small
polygon with vertices is not known when . In this paper, we
construct, for each and , a small -gon whose area is the
maximal value of a one-variable function. We show that, for all even ,
the area obtained improves by that of the best prior small -gon
constructed by Mossinghoff. In particular, for , the small -gon
constructed has maximal area.Comment: arXiv admin note: text overlap with arXiv:2009.0789
A sixteen-relator presentation of an infinite hyperbolic Kazhdan group
We provide an explicit presentation of an infinite hyperbolic Kazhdan group
with generators and relators of length at most . That group acts
properly and cocompactly on a hyperbolic triangle building of type .
We also point out a variation of the construction that yields examples of
lattices in -buildings admitting non-Desarguesian residues of
arbitrary prime power order.Comment: 9 pages, 1 figur
Largest small polygons: A sequential convex optimization approach
A small polygon is a polygon of unit diameter. The maximal area of a small
polygon with vertices is not known when . Finding the largest
small -gon for a given number can be formulated as a nonconvex
quadratically constrained quadratic optimization problem. We propose to solve
this problem with a sequential convex optimization approach, which is a ascent
algorithm guaranteeing convergence to a locally optimal solution. Numerical
experiments on polygons with up to sides suggest that optimal solutions
obtained are near-global. Indeed, for even , the algorithm
proposed in this work converges to known global optimal solutions found in the
literature
An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons
In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included
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