Existence of APAV(q,k) with q a prime power ≡5(mod8) and k≡1(mod4)

AbstractStinson introduced authentication perpendicular arrays APAλ(t,k,v), as a special kind of perpendicular arrays, to construct authentication and secrecy codes. Ge and Zhu introduced APAV(q,k) to study APA1(2,k,v) for k=5, 7. Chen and Zhu determined the existence of APAV(q,k) with q a prime power ≡3(mod4) and odd k>1. In this article, we show that for any prime power q≡5(mod8) and any k≡1(mod4) there exists an APAV(q,k) whenever q>((E+E2+4F)/2)2, where E=[(7k−23)m+3]25m−3, F=m(2m+1)(k−3)25m and m=(k−1)/4

On zero sets in Fock spaces

We prove that zero sets for distinct Fock spaces are not the same, this is an answer of a question asked by K. Zhu in \cite[Page. 209]{Zhu}

Higher level twisted Zhu algebras

The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper we consider the general set-up of a vertex algebra $V$, graded by \G/\Z for some subgroup \G of $\R$ containing $\Z$, and with a Hamiltonian operator $H$ having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level $p$ Zhu algebras \zhu_{p, \G}(V), and we prove the following theorems: For each $p$ there is a bijection between the irreducible \zhu_{p, \G}(V)-modules and the irreducible \G-twisted positive energy $V$-modules, and $V$ is (\G, H)-rational if and only if all its Zhu algebras \zhu_{p, \G}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for $H$. We provide an explicit description of the level $p$ Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \vir^c and the universal affine Kac-Moody vertex algebra V^k(\g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.Comment: 47 pages, no figure

Chip games and paintability

We prove that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$. That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with $\Theta(2^k )$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games