94 research outputs found
Groups of Fibonacci type revisited
This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway?s Fibonacci groups, the Sieradski groups, and the Gilbert-Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repov?s, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labelled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups
Modeling of internal tides in fjords
A previous model for the distribution of internal tides above irregular topography is generalized to include arbitrary stratification and a radiation condition at the open boundary. Thanks to a small amount of dissipation, this model remains valid in the presence of resonant internal tides, leading to intense wave-energy beams. An application to a Norwegian fjord correctly reproduces the observed energy pattern consisting of two beams both originating at the 60-meter deep entrance sill and extending in-fjord, one upward toward the surface, the other downward toward the bottom. After correction for the varying width of the fjord, the observed and modelled energy levels are in good agreement, especially in the upper levels where energy is the greatest. Furthermore, the substantial phase lag between these two energy beams revealed by the observations is correctly reproduced by the model. Finally, a third and very narrow energy spike is noted in the model at the level of a secondary bump inward of the sill. This beam is missed by the current meter data, because the current meters were placed only at a few selected depths. But an examination of the salinity profiles reveals a mixed layer at approximately the same depth. The explanation is that high-wave energy leads to wave breaking and vigorous mixing. The model\u27s greatest advantage is to provide the internal-tide energy distribution throughout the fjord. Discrepancies between observations and model are attributed to coarse vertical resolution in the vicinity of the sill and to unaccounted cross-fjord variations
On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs
We show that the maximum cardinality of an anti-chain composed of
intersections of a given set of n points in the plane with half-planes is close
to quadratic in n. We approach this problem by establishing the equivalence
with the problem of the maximum monotone path in an arrangement of n lines. For
a related problem on antichains in families of convex pseudo-discs we can
establish the precise asymptotic bound: it is quadratic in n. The sets in such
a family are characterized as intersections of a given set of n points with
convex sets, such that the difference between the convex hulls of any two sets
is nonempty and connected.Comment: 10 pages, 3 figures. revised version correctly attributes the idea of
Section 3 to Tverberg; and replaced k-sets by "linearly separable sets" in
the paper and the title. Accepted for publication in Israel Journal of
Mathematic
A Tverberg type theorem for matroids
Let b(M) denote the maximal number of disjoint bases in a matroid M. It is
shown that if M is a matroid of rank d+1, then for any continuous map f from
the matroidal complex M into the d-dimensional Euclidean space there exist t
\geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M
such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A
Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by
Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by
Springe
Approximating Tverberg Points in Linear Time for Any Fixed Dimension
Let P be a d-dimensional n-point set. A Tverberg-partition of P is a
partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1),
..., conv(P_r) have non-empty intersection. A point in the intersection of the
conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by
Tverberg implies that there always exists a Tverberg partition of size n/(d+1),
but it is not known how to find such a partition in polynomial time. Therefore,
approximate solutions are of interest.
We describe a deterministic algorithm that finds a Tverberg partition of size
n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we
can compute an approximate Tverberg point (and hence also an approximate
centerpoint) in linear time. Our algorithm is obtained by combining a novel
lifting approach with a recent result by Miller and Sheehy (2010).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 2012.
This version removes an incorrect example at the end of Section 3.
Halving Balls in Deterministic Linear Time
Let \D be a set of pairwise disjoint unit balls in and the
set of their center points. A hyperplane \Hy is an \emph{-separator} for
\D if each closed halfspace bounded by \Hy contains at least points
from . This generalizes the notion of halving hyperplanes, which correspond
to -separators. The analogous notion for point sets has been well studied.
Separators have various applications, for instance, in divide-and-conquer
schemes. In such a scheme any ball that is intersected by the separating
hyperplane may still interact with both sides of the partition. Therefore it is
desirable that the separating hyperplane intersects a small number of balls
only. We present three deterministic algorithms to bisect or approximately
bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present
a simple linear-time algorithm to construct an -separator for balls
in , for any , that intersects at most
balls, for some constant that depends on and . The number of
intersected balls is best possible up to the constant . Secondly, we present
a near-linear time algorithm to construct an -separator in
that intersects balls. Finally, we give a linear-time algorithm to
construct a halving line in that intersects
disks.
Our results improve the runtime of a disk sliding algorithm by Bereg,
Dumitrescu and Pach. In addition, our results improve and derandomize an
algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to
construct an onion (convex layer) decomposition for imprecise points (any point
resides at an unknown location within a given disk)
Analogues of the central point theorem for families with -intersection property in
In this paper we consider families of compact convex sets in
such that any subfamily of size at most has a nonempty intersection. We
prove some analogues of the central point theorem and Tverberg's theorem for
such families
Quantum Channels with Memory
We present a general model for quantum channels with memory, and show that it
is sufficiently general to encompass all causal automata: any quantum process
in which outputs up to some time t do not depend on inputs at times t' > t can
be decomposed into a concatenated memory channel. We then examine and present
different physical setups in which channels with memory may be operated for the
transfer of (private) classical and quantum information. These include setups
in which either the receiver or a malicious third party have control of the
initializing memory. We introduce classical and quantum channel capacities for
these settings, and give several examples to show that they may or may not
coincide. Entropic upper bounds on the various channel capacities are given.
For forgetful quantum channels, in which the effect of the initializing memory
dies out as time increases, coding theorems are presented to show that these
bounds may be saturated. Forgetful quantum channels are shown to be open and
dense in the set of quantum memory channels.Comment: 21 pages with 5 EPS figures. V2: Presentation clarified, references
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