Joshua N. Cooper

Quasirandom permutations

Chung and Graham (J. Combin. Theory Ser. A 61 (1992) 64) define quasirandom subsets of Zn to be those with any one of a large collection of equivalent random-like properties. We weaken their definition and call a subset of Zn e-balanced if its discrepancy on each interval is bounded by en. A quasirandom permutation, then, is one which maps each interval to a highly balanced set. In the spirit of previous studies of quasirandomness, we exhibit several random-like properties which are equivalent to this one, including the property of containing (approximately) the expected number of subsequences of each order-type. We present a construction for a family of strongly quasirandom permutations, and prove that this construction is essentially optimal, using a result of Schmidt on the discrepancy of sequences of real numbers.

Asymmetric binary covering codes

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Qn such that every vector x ∈ Qn can be obtained from some vector c ∈ c by changing at most R 1s of c to 0s, where R is as small as possible. K+ (n, R) is defined as the smallest size of such a code. We show K+(n, R) ∈ Θ (2n/nR) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K+(n,n -R) = R+ 1 for constant coradius R iff n ≥ R(R + 1)/2. These two results are extendetd to near-constant R and R, respectively. Various bounds on K+ are given in terms of the total number of 0s or 1s in a minimal code. The dimension of a minimal asymmetric linear binary code ([n, R]+-code) is determined to be min{0, n - R}. We conclude by discussing open problems and techniques to compute explicit values for K+, giving a table of best-known bounds.

Analytic connectivity of k-uniform hypergraphs

In this paper, we study the analytic connectivity of a k-uniform hypergraph H, denoted by . In addition to computing the analytic connectivity of a complete k-graph, we present several bounds on analytic connectivity that relate it with other graph invariants, such as degree, vertex connectivity, diameter and isoperimetric number.

Computing hypermatrix spectra with the Poisson product formula

We compute the spectrum of the ‘all ones’ hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalues’ multiplicities, a seemingly elusive aspect of the spectral theory of tensors. We also give a distributional picture of the spectrum as a point-set in the complex plane. Finally, we use the technique to analyse the spectrum of ‘sunflower hypergraphs’, a class that has played a prominent role in extremal hypergraph theory.

Uniquely C 4 -Saturated Graphs

For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H, but the addition of any edge from

Reciprocals of Binary Power Series

{\overline{G}}

Bioterrorism and the Fermi Paradox

to G completes exactly one copy of H. Using a combination of algebraic methods and counting arguments, we determine all the uniquely C4-saturated graphs; there are only ten of them.

Monochromatic Boxes in Colored Grids

Abstract We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue–Morse sequence if they greedily demand their chances to fire as soon as the others a priori probability of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions, which says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the Thue–Morse sequence in terms of certain expansions in fractional bases close to, but greater than, 1.