Structures and lower bounds for binary covering arrays

A $q$-ary $t$-covering array is an $m\times n$ matrix with entries from $\{0,1,\dots ,q?1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for given $t$ and $n$, or maximize $n$ for given $t$ and $m$. For $t=2$ and $q=2$, it is completely solved by Rényi, Katona, and Kleitman and Spencer. They also show that maximal binary 2-covering arrays are uniquely determined. Roux found a lower bound of $m$ for a general $t,n$, and $q$. In this article, we show that $m\times n$ binary 2-covering arrays under some constraints on $m$ and $n$ come from the maximal covering arrays. We also improve the lower bound of Roux for $t=3$ and $q=2$, and show that some binary 3 or 4-covering arrays are uniquely determined.     

On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators $f?w?(f°\phi )$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).     

A new class of n-ary hyperoperations

To a given $n$-ary hyperoperation $f$ on a universe $H$ and a unary hyperoperation $\phi$ on $H$ we define a new $n$-ary hyperoperation ${\partial }_f^{\phi }$ on $H$. We study the associativity, weak associativity and reproductivity of $n$-ary hyperoperations ${\partial }_f^{\phi }$.     

Beau bounds for multicritical circle maps

Let $f:S^1\to S^1$ be a $C^3$ homeomorphism without periodic points having a finite number of critical points of power-law type. In this paper we establish real a-priori bounds, on the geometry of orbits of $f$, which are beau in the sense of Sullivan, i.e. bounds that are asymptotically universal at small scales. The proof of the beau bounds presented here is an adaptation, to the multicritical setting, of the one given by the second author and de Melo in de Faria and de Melo (1999), for the case of a single critical point.     

On the independence number of the power graph of a finite group

The power graph ${\Gamma }_G$ of a finite group $G$ is the graph whose vertex set is $G$, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ${\Gamma }_G$ and characterize the groups achieving the bounds. Moreover, we determine the independence number of ${\Gamma }_G$ if $G$ is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups $G$ whose power graphs have independence number 3 or $n?2$, where $n$ is the order of $G$.