Hamiltonicity of the Pyramid Network with or without Fault
Sarbazi-Azad, Ould-Khaoua, and Mackenzie proved in 2001 that there exists a Hamiltonian cycle in a pyramid network and they also constructed a Hamiltonian path between apex and each of 4 frontiers of a pyramid network. The fault tolerance is a crucial matter for parallel computing, especially in a large network. This work improves Sarbazi-Azad et al.'s result and considers other relative problems in pyramid networks such as the fault tolerant Hamiltonian problem and the Hamiltonian-connected problem. The problem of finding Hamiltonian cycles in a pyramid network with one faulty node (link) is investigated. Additionally, the Hamiltonian-connectedness of a pyramid network can be shown by constructing a Hamiltonian path between any two distinct nodes in it.
We study in perturbative QCD the helicity amplitudes of the process ${\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{p}\ensuremath{\rho}p$ at large virtualities Q of the photon ${\ensuremath{\gamma}}^{*}.$ We estimate all spin-flip amplitudes taking into account an important effect of the scale behavior of the gluon density. The transition of a transverse virtual photon to a longitudinal vector meson is not small at typical conditions at the DESY ep collider HERA. This helicity nonconserving amplitude leads by interference to a measurable effect in the distribution of the angle between the electron scattering and the meson production planes.