32,235 research outputs found
On lattices of convex sets in R^n
Properties of several sorts of lattices of convex subsets of R^n are
examined. The lattice of convex sets containing the origin turns out, for n>1,
to satisfy a set of identities strictly between those of the lattice of all
convex subsets of R^n and the lattice of all convex subsets of R^{n-1}. The
lattices of arbitrary, of open bounded, and of compact convex sets in R^n all
satisfy the same identities, but the last of these is join-semidistributive,
while for n>1 the first two are not. The lattice of relatively convex subsets
of a fixed set S \subseteq R^n satisfies some, but in general not all of the
identities of the lattice of ``genuine'' convex subsets of R^n.Comment: 35 pages, to appear in Algebra Universalis, Ivan Rival memorial
issue. See also http://math.berkeley.edu/~gbergman/paper
Minimal genera of open 4-manifolds
We study exotic smoothings of open 4-manifolds using the minimal genus
function and its analog for end homology. While traditional techniques in open
4-manifold smoothing theory give no control of minimal genera, we make progress
by using the adjunction inequality for Stein surfaces. Smoothings can be
constructed with much more control of these genus functions than the compact
setting seems to allow. As an application, we expand the range of 4-manifolds
known to have exotic smoothings (up to diffeomorphism). For example, every
2-handlebody interior (possibly infinite or nonorientable) has an exotic
smoothing, and "most" have infinitely, or sometimes uncountably many,
distinguished by the genus function and admitting Stein structures when
orientable. Manifolds with 3-homology are also accessible. We investigate
topological submanifolds of smooth 4-manifolds. Every domain of holomorphy
(Stein open subset) in the complex plane is topologically isotopic to
uncountably many other diffeomorphism types of domains of holomorphy with the
same genus functions, or with varying but controlled genus functions.Comment: 30 pages, 1 figure. v3 is essentially the version published in
Geometry and Topology, obtained from v2 by major streamlining for
readability. Several new examples added since v2; see last paragraph of
introduction for detail
High-Rate Regenerating Codes Through Layering
In this paper, we provide explicit constructions for a class of exact-repair
regenerating codes that possess a layered structure. These regenerating codes
correspond to interior points on the storage-repair-bandwidth tradeoff, and
compare very well in comparison to scheme that employs space-sharing between
MSR and MBR codes. For the parameter set with , we
construct a class of codes with an auxiliary parameter , referred to as
canonical codes. With in the range , these codes operate in
the region between the MSR point and the MBR point, and perform significantly
better than the space-sharing line. They only require a field size greater than
. For the case of , canonical codes can also be shown to
achieve an interior point on the line-segment joining the MSR point and the
next point of slope-discontinuity on the storage-repair-bandwidth tradeoff.
Thus we establish the existence of exact-repair codes on a point other than the
MSR and the MBR point on the storage-repair-bandwidth tradeoff. We also
construct layered regenerating codes for general parameter set ,
which we refer to as non-canonical codes. These codes also perform
significantly better than the space-sharing line, though they require a
significantly higher field size. All the codes constructed in this paper are
high-rate, can repair multiple node-failures and do not require any computation
at the helper nodes. We also construct optimal codes with locality in which the
local codes are layered regenerating codes.Comment: 20 pages, 9 figure
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
Subset Warping: Rubber Sheeting with Cuts
Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself
Dimers, webs, and positroids
We study the dimer model for a planar bipartite graph N embedded in a disk,
with boundary vertices on the boundary of the disk. Counting dimer
configurations with specified boundary conditions gives a point in the totally
nonnegative Grassmannian. Considering pairing probabilities for the
double-dimer model gives rise to Grassmann analogues of Rhoades and Skandera's
Temperley-Lieb immanants. The same problem for the (probably novel)
triple-dimer model gives rise to the combinatorics of Kuperberg's webs and
Grassmann analogues of Pylyavskyy's web immanants. This draws a connection
between the square move of plabic graphs (or urban renewal of planar bipartite
graphs), and Kuperberg's square reduction of webs. Our results also suggest
that canonical-like bases might be applied to the dimer model.
We furthermore show that these functions on the Grassmannian are compatible
with restriction to positroid varieties. Namely, our construction gives bases
for the degree two and degree three components of the homogeneous coordinate
ring of a positroid variety that are compatible with the cyclic group action.Comment: 25 page
Subset currents on free groups
We introduce and study the space of \emph{subset currents} on the free group
. A subset current on is a positive -invariant locally finite
Borel measure on the space of all closed subsets of consisting of at least two points. While ordinary geodesic currents
generalize conjugacy classes of nontrivial group elements, a subset current is
a measure-theoretic generalization of the conjugacy class of a nontrivial
finitely generated subgroup in , and, more generally, in a word-hyperbolic
group. The concept of a subset current is related to the notion of an
"invariant random subgroup" with respect to some conjugacy-invariant
probability measure on the space of closed subgroups of a topological group. If
we fix a free basis of , a subset current may also be viewed as an
-invariant measure on a "branching" analog of the geodesic flow space for
, whose elements are infinite subtrees (rather than just geodesic lines)
of the Cayley graph of with respect to .Comment: updated version; to appear in Geometriae Dedicat
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