The rotation distance d(S, T ) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T . While it is known that d(S, T ) ≤ 2n− 6, a well known conjecture states that there are trees for which this bound is sharp for any value of n ≥ 11. We are unable to prove the conjecture, but we give here some simple criteria for lower bound evaluation, leading fo...

For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = $\cap$ [0, -- 1], set W (S) e|L| c. In 1978 Wilf asked, in equivalent terms, whether $\ge$ 0 always holds, question known since as Wilf's conjecture. Using closely related lower bound $\le$ (S), we show that if |L| 12 then 0, thereby settling conjecture this case. This is best possible, cases are...

We consider the analogue of Hurwitz curves, smooth projective curves C of genus g ≥ 2 that realize equality in the Hurwitz bound |Aut(C)| ≤ 84(g − 1), to smooth compact quotients S of the unit ball in C. When S is arithmetic, we show that |Aut(S)| ≤ 288e(S), where e(S) is the (topological) Euler characteristic, and in the case of equality show that S is a regular cover of a particular Deligne–M...

The critical number of G, denoted by c(G), is the smallest s such that Σ(S) = G for every subset S of G with cardinality s not containing 0. The parameter c(G) was first studied by Erdős and Heilbronn in [4]. They obtained the inequality c(Zp) ≤ 3 √ 6p. Olson proved in [13] that c(Zp) ≤ √ 4p− 3 + 1. The authors of [1] obtained the inequality c(Zp) ≤ √ 4p− 7. The evaluation of c(G) for groups wi...

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I. S. Luthar and I. B. S. Passi which sometimes permits an answer to this conjecture. We illustrate the method on certain explicit examples. We prove wit...

The Sensitivity Conjecture, posed in 1994, states that the fundamental measures known as the sensitivity and block sensitivity of a Boolean function f , s(f) and bs(f) respectively, are polynomially related. It is known that bs(f) is polynomially related to important measures in computer science including the decision-tree depth, polynomial degree, and parallel RAM computation time of f , but l...

Let X = (xij) and Y = (yij) be generic n by n matrices and Z = XY − Y X. Let S = k[x11, . . . , xnn, y11, . . . , ynn], where k is a field, let I be the ideal generated by the entries of Z and let R = S/I . We give a conjecture on the first syzygies of I , show how these can be used to give a conjecture on the canonical module of R. Using this and the Hilbert series of I we give a conjecture on...