O culodentodigital syndrome (ODD; OMIM 164200) is a congenital disorder characterised by developmental abnormalities of the face, eyes, limbs, and dentition. ODD is inherited in an autosomal dominant fashion and displays high penetrance but variable expression. In addition, a high rate of de novo mutations is observed. Facially, affected patients exhibit a long, narrow nose with hypoplastic ala...

Strong starters and skew starters have been widely used in various combinatorial designs. In particular skew balanced starters and symmetric skew balanced starters are crucially used in the construction of completely balanced Howell rotations. Let n = 2mfc + 1 be an odd prime power where m > 2 and k is an odd number. The existence of symmetric skew balanced starters for GF(n) has been proved fo...

Complete next-to-leading (chiral) order (NLO) expressions for the strong isospin-breaking (IB) contributions in K → ππ are used to discuss (1) for CP-even, the impact on the magnitude of the ∆I = 1/2 Rule, and (2) for CP-odd, the strong IB correction, Ωst, for the gluonic penguin contribution to ǫ /ǫ, with particular emphasis on the strong low-energy constant (LEC) and loop contributions, numer...

Krein [1] mentioned that for each PD equation we have two extreme operators, one is the minimal in which solution and its derivatives on the boundary are zero, the other one is the maximal operator in which there is no prescribed boundary conditions. They claim it is not possible to have a related boundary value problem for an arbitrarily chosen operator in between. They have only considered lo...

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|). This is independent of the recent proof of the strong perfect graph conjecture.

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge’s famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (”odd holes”) and their complements (”odd antiholes”). Only two constructions (due to Chvátal, Graham, Pero...

We study the linear span of commutators free random variables and show that these are only quadratic forms which satisfy following equivalent properties: preservation infinite divisibility, strong cancellation odd cumulants, symmetric distribution for any family.