10,019 research outputs found
On Point Sets in Vector Spaces over Finite Fields That Determine Only Acute Angle Triangles
For three points , and in the -dimensional
space \F_q^n over the finite field \F_q of elements we give a natural
interpretation of an acute angle triangle defined by this points. We obtain an
upper bound on the size of a set \cZ such that all triples of distinct points
\vec{u}, \vec{v}, \vec{w} \in \cZ define acute angle triangles. A similar
question in the real space \cR^n dates back to P. Erd{\H o}s and has been
studied by several authors
Polygons of the Lorentzian plane and spherical simplexes
It is known that the space of convex polygons in the Euclidean plane with
fixed normals, up to homotheties and translations, endowed with the area form,
is isometric to a hyperbolic polyhedron. In this note we show a class of convex
polygons in the Lorentzian plane such that their moduli space, if the normals
are fixed and endowed with a suitable area, is isometric to a spherical
polyhedron. These polygons have an infinite number of vertices, are space-like,
contained in the future cone of the origin, and setwise invariant under the
action of a linear isometry.Comment: New text, title slightly change
Cone fields and topological sampling in manifolds with bounded curvature
Often noisy point clouds are given as an approximation of a particular
compact set of interest. A finite point cloud is a compact set. This paper
proves a reconstruction theorem which gives a sufficient condition, as a bound
on the Hausdorff distance between two compact sets, for when certain offsets of
these two sets are homotopic in terms of the absence of {\mu}-critical points
in an annular region. Since an offset of a set deformation retracts to the set
itself provided that there are no critical points of the distance function
nearby, we can use this theorem to show when the offset of a point cloud is
homotopy equivalent to the set it is sampled from. The ambient space can be any
Riemannian manifold but we focus on ambient manifolds which have nowhere
negative curvature. In the process, we prove stability theorems for
{\mu}-critical points when the ambient space is a manifold.Comment: 20 pages, 3 figure
Hyperbolic Dehn filling in dimension four
We introduce and study some deformations of complete finite-volume hyperbolic
four-manifolds that may be interpreted as four-dimensional analogues of
Thurston's hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone
four-manifolds that interpolates between two hyperbolic four-manifolds
and with the same volume . The deformation looks
like the familiar hyperbolic Dehn filling paths that occur in dimension three,
where the cone angle of a core simple closed geodesic varies monotonically from
to . Here, the singularity of is an immersed geodesic surface
whose cone angles also vary monotonically from to . When a cone angle
tends to a small core surface (a torus or Klein bottle) is drilled
producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise,
including one case where a degeneration occurs when the cone angles tend to
, like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional
deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio
Topological robotics: motion planning in projective spaces
We study an elementary problem of topological robotics: rotation of a line,
which is fixed by a revolving joint at a base point: one wants to bring the
line from its initial position to a final position by a continuous motion in
the space. The final goal is to construct an algorithm which will perform this
task once the initial and final positions are given.
Any such motion planning algorithm will have instabilities, which are caused
by topological reasons. A general approach to study instabilities of robot
motion was suggested recently by the first named author. With any
path-connected topological space X one associates a number TC(X), called the
topological complexity of X. This number is of fundamental importance for the
motion planning problem: TC(X) determines character of instabilities which have
all motion planning algorithms in X.
In the present paper we study the topological complexity of real projective
spaces. In particular we compute TC(RP^n) for all n<24. Our main result is that
(for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is
equivalent to finding the smallest k such that RP^n can be immersed into the
Euclidean space R^{k-1}.Comment: 16 page
Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient
We study the geometry of domains in complete metric measure spaces equipped
with a doubling measure supporting a -Poincar\'e inequality. We propose a
notion of \emph{domain with boundary of positive mean curvature} and prove
that, for such domains, there is always a solution to the Dirichlet problem for
least gradients with continuous boundary data. Here \emph{least gradient} is
defined as minimizing total variation (in the sense of BV functions) and
boundary conditions are satisfied in the sense that the \emph{boundary trace}
of the solution exists and agrees with the given boundary data. This extends
the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via
counterexamples we also show that uniqueness of solutions and existence of
\emph{continuous} solutions can fail, even in the weighted Euclidean setting
with Lipschitz weights
Collapsibility of CAT(0) spaces
Collapsibility is a combinatorial strengthening of contractibility. We relate
this property to metric geometry by proving the collapsibility of any complex
that is CAT(0) with a metric for which all vertex stars are convex. This
strengthens and generalizes a result by Crowley. Further consequences of our
work are:
(1) All CAT(0) cube complexes are collapsible.
(2) Any triangulated manifold admits a CAT(0) metric if and only if it admits
collapsible triangulations.
(3) All contractible d-manifolds () admit collapsible CAT(0)
triangulations. This discretizes a classical result by Ancel--Guilbault.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes
has been removed and forms a new paper, called "Barycentric subdivisions of
convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration
of manifolds has also been removed and forms now a third paper, called "A
Cheeger-type exponential bound for the number of triangulated manifolds"
(arXiv:1710.00130
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