365 research outputs found
A coding problem for pairs of subsets
Let be an --element finite set, an integer. Suppose that
and are pairs of disjoint -element subsets of
(that is, , , ). Define the distance of these pairs by . This is the
minimum number of elements of one has to move to obtain the other
pair . Let be the maximum size of a family of pairs of
disjoint subsets, such that the distance of any two pairs is at least .
Here we establish a conjecture of Brightwell and Katona concerning an
asymptotic formula for for are fixed and . Also,
we find the exact value of in an infinite number of cases, by using
special difference sets of integers. Finally, the questions discussed above are
put into a more general context and a number of coding theory type problems are
proposed.Comment: 11 pages (minor changes, and new citations added
Two-part and k-Sperner families: New proofs using permutations
This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. ErdĆs and G. O. H. Katona, J. Combin. Theory. Ser. A,4
Existence of a maximum balanced matching in the hypercube
We prove, that for the maximum possible edges can be chosen simultaneously from each parallel class of the n-cube in such a way that no two edges have a common vertex. © 2013 Copyright Grace Scientific Publishing, LLC
The optical system of the H.E.S.S. imaging atmospheric Cherenkov telescopes, Part II: mirror alignment and point spread function
Mirror facets of the H.E.S.S. imaging atmospheric Cherenkov telescopes are
aligned using stars imaged onto the closed lid of the PMT camera, viewed by a
CCD camera. The alignment procedure works reliably and includes the automatic
analysis of CCD images and control of the facet alignment actuators. On-axis,
80% of the reflected light is contained in a circle of less than 1 mrad
diameter. The spot widens with increasing angle to the telescope axis. In
accordance with simulations, the spot size has roughly doubled at an angle of
1.4 degr. from the axis. The expected variation of spot size with elevation due
to deformations of the support structure is visible, but is completely
non-critical over the usual working range. Overall, the optical quality of the
telescope exceeds the specifications.Comment: 23 pages, 13 figure
ADAPTIVE MAJORITY PROBLEMS FOR RESTRICTED QUERY GRAPHS AND FOR WEIGHTED SETS
Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))
The optical system of the H.E.S.S. imaging atmospheric Cherenkov telescopes, Part I: layout and components of the system
H.E.S.S. -- the High Energy Stereoscopic System -- is a new system of large
imaging atmospheric Cherenkov telescopes, with about 100 m^2 mirror area for
each of four telescopes, and photomultiplier cameras with a large field of view
(5 degr.) and small pixels (0.16 degr.). The dish and reflector are designed to
provide good imaging properties over the full field of view, combined with
mechanical stability. The paper describes the design criteria and
specifications of the system, and the individual components -- dish, mirrors,
and Winston cones -- as well as their characteristics. The optical performance
of the telescope as a whole is the subject of a companion paper.Comment: 28 pages, 20 figure
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Regular dendritic patterns induced by non-local time-periodic forcing
The dynamic response of dendritic solidification to spatially homogeneous
time-periodic forcing has been studied. Phase-field calculations performed in
two dimensions (2D) and experiments on thin (quasi 2D) liquid crystal layers
show that the frequency of dendritic side-branching can be tuned by oscillatory
pressure or heating. The sensitivity of this phenomenon to the relevant
parameters, the frequency and amplitude of the modulation, the initial
undercooling and the anisotropies of the interfacial free energy and molecule
attachment kinetics, has been explored. It has been demonstrated that besides
the side-branching mode synchronous with external forcing as emerging from the
linear Wentzel-Kramers-Brillouin analysis, modes that oscillate with higher
harmonic frequencies are also present with perceptible amplitudes.Comment: 15 pages, 23 figures, Submitted to Phys. Rev.
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