A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ 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. Erdős 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 $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ 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 $S_A$ 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 $A$ 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.