Generalized solution for the Herman Protocol Conjecture

We have a cycle of $N$ nodes and there is a token on an odd number of nodes. At each step, each token independently moves to its clockwise neighbor or stays at its position with probability $\frac{1}{2}$. If two tokens arrive to the same node, then we remove both of them. The process ends when only one token remains. The question is that for a fixed $N$, which is the initial configuration that maximizes the expected number of steps $E(T)$. The Herman Protocol Conjecture says that the $3$-token configuration with distances $\lfloor\frac{N}{3}\rfloor$ and $\lceil\frac{N}{3}\rceil$ maximizes $E(T)$. We present a proof of this conjecture not only for $E(T)$ but also for $E\big(f(T)\big)$ for some function $f:\mathbb{N}\rightarrow\mathbb{R}^{+}$ which method applies for different generalizations of the problem

Hydrothermal Synthesis of Sr-Doped Hydroxyapatite and Its Antibacterial Activity

In this study, we prepared hydroxyapatite (HAP) samples using hydrothermal method. We investigated the effect of reaction conditions such as phosphate excess applying 1.49 and 1.67 (as stoichiometric) Ca/P ratio, pH (9/ 10/ 11/ 12) and time (4/ 8/ 12/ 24 h). Sample characterization was carried out by XRD and SEM. The results showed, all samples had HAP structure, however, lower Ca/P ratio, larger reaction time and setting the pH to 10 increased the crystallinity. Then, we synthetized Sr-doped HAP samples, varying the Sr concentration using 2, 4, 6, 8 and 10 % Sr/ (Ca+Sr). The Sr content was revealed by EDX. Sr-incorporation did not change the obtained crystalline HAP phase but the unit cell parameters increased. We calculated lattice constants and found that a, b changed from 9.4310 Å to 9.4700 Å, c from 6.8819 Å to 6.9227 Å and the unit cell volume from 530.0951 Å3 to 537.6556 Å3 due to the larger ionic radius of Sr compared to Ca. The pure and doped samples had uniform, mostly needle-like morphology with 100-300 nm length and 25-100 nm. In vitro cytotoxicity tests revealed evident antibacterial activity in the case of doped samples compared to pure HAP against E. coli

Magyar Tanítóképző 10 (1895) 09

Magyar Tanítóképző A Tanítóképző-intézeti Tanárok Országos Egyesületének közlönye 10. évfolyam, 09. füzet Budapest, 1895. november h