A Simplex-Type Algorithm for General Transportation Problems: An Alternative to Stepping-Stone

This paper re-examines use of the linear programming (LP) formulation to solve the transportation problem (TP). The proposed method is a general-purpose algorithm which uses only one operation, the Gauss Jordan pivoting used in the simplex method. The final tableau can be used for post-optimality analysis of TP. This algorithm appears to be faster than simplex, more general than stepping-stone and simpler than both in solving general TP. A numerical example illustrates the methodology. It is assumed the reader is familiar with simplex terminology.     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Low temperature magnetic susceptibility and exchange interaction in Cs3Cr2Cl9

Magnetic susceptibility of Cs₃Cr₂Cl₉ as a single crystal is studied in the temperature range 4.2–77 K. A maximum is obtained at 25 ± 1 K. These experimental data are interpreted by considering the isotropic exchange interaction between two spin quadruplets. The exchange constant J is found to be equal to - 13 cm^?1.     

Dinuclear Complexes with Predictable Magnetic Properties

When two paramagnetic transition metal ions are present in the same molecular entity, the magnetic properties can be totally different from the sum of the magnetic properties of each ion surrounded by its nearest neighbors. These new properties depend on the nature and the magnitude of the interaction between the metal ions through the bridging ligands. If both ions have an unpaired electron (e.g. Cu²⁺ ions), then the molecular state of lowest energy is either a spin singlet or a spin triplet. In the former case, the interaction is said to be antiferromagnetic, in the latter case ferromagnetic. The nature and the order of magnitude of the interaction can be engineered by judiciously choosing the interacting metal ions and the bridging and terminal ligands, and, thus, by the symmetry and the delocalization of the orbitals centered on the metal ions and occupied by the unpaired electrons (magnetic orbitals). The first success in this “molecular engineering” of bimetallic compounds was in the synthesis of a Cu²⁺VO²⁺ heterobimetallic complex in which the interaction is purely ferro-magnetic. The same strategy could be utilized for designing molecular ferromagnets, one of the major challenges in the area of molecular materials. Another striking result is the possibility of tuning the magnitude of the interaction through a given bridging network by modifying the nature of the terminal ligands, which, in some way, play the role of “adjusting screws”. By careful selection of the bridging and terminal ligands, a very large antiferro-magnetic interaction can be achieved, even if the metal ions are far away from each other. Some sulfur-containing bridges are especially suitable in this respect.