Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

Josephson current through a half metal at low temperatures

We study the Josephson effect in the superconductor/diffusive half metal/superconductor junctions by using the recursive Green function method. In the presence of spin-flip scatterings at the interface, odd-frequency spin-triplet Cooper pairs penetrate deeply into a half metal and carry Josephson current. The critical Josephson current increases with decreasing temperatures near the transition temperature. At low temperatures, however, the critical current decreases with decreasing temperatures. Such reentrant behavior is unusual in the case of s-wave superconductor junctions. The penetration of odd-frequency pairs modifies quasiparticle density of states in a half metal near the Fermi energy, which is responsible for the nonmonotonic temperature dependence of critical Josephson current.     

Ultrasound-assisted green syntheses of novel pyrimidine derivatives and their comparison with conventional methods

Ultrasound-assisted green syntheses of novel potentially bioactive pyrimidine derivatives have been carried out. The same compounds were obtained by conventional methods of synthesis, and the reaction times and yields of final products obtained by these two methods were compared. It was found that the time of utrasound-promoted reactions was reduced by almost 6–96 times, and their yields were equal or turn out to be greater compared to the traditional approach. The synthesized compounds showed a pronounced stimulating effect on plant growth. The most active derivatives were selected for deeper biological studies and subsequent field trials.