139,218 research outputs found
On building 4-critical plane and projective plane multiwheels from odd wheels
We build unbounded classes of plane and projective plane multiwheels that are
4-critical that are received summing odd wheels as edge sums modulo two. These
classes can be considered as ascending from single common graph that can be
received as edge sum modulo two of the octahedron graph O and the minimal wheel
W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among
which are those that quadrangulate projective plane, i.e., graphs from
Gr\"otzsch class, received applying Mycielski's Construction to odd cycle.Comment: 10 page
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
External grind-hardening forces modelling and experimentation
Grind hardening process utilizes the heat generated in the grinding area for the surface heat treatment of the workpiece. The workpiece surface is heated above the austenitizing temperature by using large values of depth of cut and low workpiece feed speeds. However, such process parameter combinations result in high process forces that inhibit the broad application of grind hardening to smaller grinding machines. In the present paper, modelling and predicting of the process forces as a function of the process parameters are presented. The theoretical predictions present good agreement with experimental results. The results of the study can be used for the prediction of the grind hardening process forces and, therefore, optimize the process parameters so as to be used with every size grinding machine
Two applications of elementary knot theory to Lie algebras and Vassiliev invariants
Using elementary equalities between various cables of the unknot and the Hopf
link, we prove the Wheels and Wheeling conjectures of [Bar-Natan, Garoufalidis,
Rozansky and Thurston, arXiv:q-alg/9703025] and [Deligne, letter to Bar-Natan,
January 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/], which give,
respectively, the exact Kontsevich integral of the unknot and a map
intertwining two natural products on a space of diagrams. It turns out that the
Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on
a wire), and its intertwining property is analogous to the computation of 1+1=2
on an abacus. The Wheels conjecture is proved from the fact that the k-fold
connected cover of the unknot is the unknot for all k. Along the way, we find a
formula for the invariant of the general (k,l) cable of a knot. Our results can
also be interpreted as a new proof of the multiplicativity of the
Duflo-Kirillov map S(g)-->U(g) for metrized Lie (super-)algebras g.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper1.abs.htm
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
Nearly Fuchsian surface subgroups of finite covolume Kleinian groups
Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove
that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally
conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous
work of Kahn and Markovic, this proves that every finite covolume Kleinian
group has a nearly Fuchsian surface subgroup.Comment: v2: Final prepublication versio
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