Asymmetric Binary Covering Codes

An asymmetric binary covering code of length n and radius R is a subset of the n-cube Q_n such that every vector xQ_n can be obtained from some vector c by changing at most R 1's of c to 0's, 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/n^R) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K⁺(n,n− )= +1 for constant coradius iff n ( +1)/2. These two results are extended to near-constant R and , respectively. Various bounds on K⁺ are given in terms of the total number of 0's or 1's 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.     

Closed Trapped Surfaces in Cosmology

The existence of closed trapped surfaces need not imply a cosmological singularity when the spatial hypersurfaces are compact. This is illustrated by a variety of examples, in particular de Sitter spacetime admits many closed trapped surfaces and obeys the null convergence condition but is non-singular in the k = +1 frame.     

The synthesis of functionalised chiral bicyclic lactam and lactone N-oxides using a tandem Cope elimination/reverse Cope elimination protocol

Functionalised hydroxylamine derivatives of (S)-proline and (R)-pipecolic acid have been prepared using a Cope elimination. These undergo reverse Cope elimination onto a pendant double bond to give bicyclic lactam and lactone N-oxides containing three contiguous chiral centres, this extends the scope and applicability of the reverse Cope elimination in the synthesis of heterocyclic systems by incorporation of the lactone and lactam structural motifs.     

Building up a Sequence of Optimum Media Schedules

A mathematical model is proposed for finding the response to an advertising campaign. Although, hypothetically, there is no restriction on the media involved, the assumptions made are more suited to a campaign in newspapers or magazines, rather than, for example, on television. The deficiencies in the model are discussed, and the use of such a model is compared with the use of simulation techniques. Two previous attempts to optimize similar models are discussed, and a technique for building up a sequence of optimum schedules with an example and a proof of the optimum properties are presented.     

Jets and correlations in large pT reactions

Recent experimental data on correlations in large transverse momentum reactions are analyzed in a two-jet picture. Scaling in jet fragmentation is shown to account well for the existing data. The formulae that are given can be used as a basis for a more detailed analysis of the dynamics. A particular result is that we predict that a two-jet structure can also be produced in events where it is not required that any single hadron has large p_T, with a cross section as large as $\text{1}\text{10}$ mb or more (at ISR energies with the total transverse momentum of each jet ?2.5 GeV/c).     

Ordered steps on Bi(∼ 0001) vicinal surfaces by laser pulse anneal

LEED patterns characteristic of an ordered array of equally spaced steps of minimum height resulted from pulsing disordered Bi(～ 0001) vicinal surfaces. Ten XeCl eximer laser pulses at λ = 308 nm, of 10 ns duration and $\text{4 mJ}\text{cm}{}^2$ each and at 0.5 s intervals optimized ordering on the Bi(101̄16) surface, but additional exposure degraded the surface crystallinity as did a twentyfold increase in power density.