We examine the open problem of finding the shortest string that contains each of the n! permutations of n symbols as contiguous substrings (i.e., the shortest superpermutation on n symbols). It has been conjectured that the shortest superpermutation has length ∑n k=1 k! and that this string is unique up to relabelling of the symbols. We provide a construction of short superpermutations that sho...

We describe how unintegrated parton distributions can be calculated from conventional integrated distributions. We extend and improve the ‘last-step’ evolution approach, and explain why doubly-unintegrated parton distributions are necessary. We generalise ktfactorisation to (z, kt)-factorisation. We apply the formalism to inclusive jet production in deep-inelastic scattering, mainly at leading ...

The characteristic functional (c.fl.) of a doubly stochastic Poisson process (DSPP) is studied and it provides us the finite dimensional distributions of the process and so its moments. It is also studied the case of a DSPP which intensity is a narrow-band process. The Karhunen–Loève expansion of its intensity is used to obtain the probability distribution function and a decomposition of this P...

To address different uncertainties associated with patient recruitment in multicentre clinical trials a mixed Poisson model is elaborated (patients arrive at different clinical centres according to delayed doubly stochastic Poisson processes with gamma distributed rates). Analytic approach to modeling and predicting the recruitment at the initial and ongoing stages is suggested. A Bayesian appr...

Free vibration of a simply-supported magneto-electro-elastic doubly-curved nano-shell is studied based on the first-order shear deformation theory in the presence of the rotary inertia effect. To model the electric and magnetic behaviors of the nano-shell, Gauss’s laws for electrostatics and magneto statics are used. By using Navier’s method, the partial differential equations of motion are red...

We consider a class of reflected backward doubly stochastic differential equations with time delayed generator (in short RBDSDE with time delayed generator), in this case generator at time t can depend on the values of a solution in the past. Under a Lipschitz condition, we ensure the existence and uniqueness of the solution.

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighbourhood is as large as its first outneighbourhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arbor...