34,211 research outputs found
Counting d-polytopes with d+3 vertices
We completely solve the problem of enumerating combinatorially inequivalent
-dimensional polytopes with vertices. A first solution of this
problem, by Lloyd, was published in 1970. But the obtained counting formula was
not correct, as pointed out in the new edition of Gr\"unbaum's book. We both
correct the mistake of Lloyd and propose a more detailed and self-contained
solution, relying on similar preliminaries but using then a different
enumeration method involving automata. In addition, we introduce and solve the
problem of counting oriented and achiral (i.e. stable under reflection)
-polytopes with vertices. The complexity of computing tables of
coefficients of a given size is then analyzed. Finally, we derive precise
asymptotic formulas for the numbers of -polytopes, oriented -polytopes
and achiral -polytopes with vertices. This refines a first asymptotic
estimate given by Perles.Comment: 24 page
Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem
This is the second in a series of papers dedicated to studying w-knots, and
more generally, w-knotted objects (w-braids, w-tangles, etc.). These are
classes of knotted objects that are wider but weaker than their "usual"
counterparts. To get (say) w-knots from usual knots (or u-knots), one has to
allow non-planar "virtual" knot diagrams, hence enlarging the the base set of
knots. But then one imposes a new relation beyond the ordinary collection of
Reidemeister moves, called the "overcrossings commute" relation, making
w-knotted objects a bit weaker once again. Satoh studied several classes of
w-knotted objects (under the name "weakly-virtual") and has shown them to be
closely related to certain classes of knotted surfaces in R4. In this article
we study finite type invariants of w-tangles and w-trivalent graphs (also
referred to as w-tangled foams). Much as the spaces A of chord diagrams for
ordinary knotted objects are related to metrized Lie algebras, the spaces Aw of
"arrow diagrams" for w-knotted objects are related to not-necessarily-metrized
Lie algebras. Many questions concerning w-knotted objects turn out to be
equivalent to questions about Lie algebras. Most notably we find that a
homomorphic universal finite type invariant of w-foams is essentially the same
as a solution of the Kashiwara-Vergne conjecture and much of the
Alekseev-Torossian work on Drinfel'd associators and Kashiwara-Vergne can be
re-interpreted as a study of w-foams.Comment: 57 pages. Improvements to the exposition following a referee repor
Ripples and Grains Segregation on Unpaved Road
Ripples or corrugations are common phenomena observed in unpaved roads in
less developed countries or regions. They cause several damages in vehicles
leading to increased maintenance and product costs. In this paper, we present a
computational study about the so-called washboard roads. Also, we study grain
segregation on unpaved roads. Our simulations have been performed by the
Discrete Element Method (DEM). In our model, the grains are regarded as soft
disks. The grains are subjected to a gravitational field and both translational
and rotational movements are allowed. The results show that wheels' of
different sizes, weights and moving with different velocities can change
corrugations amplitude and wavelength. Our results also show that some
wavelength values are related to specific wheels' speed intervals. Segregation
has been studied in roads formed by three distinct grain diameters
distribution. We observed that the phenomenon is more evident for higher grain
size dispersion
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