Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u_t=V(u)u_x, by the generalized hodograph method requires the diagonalizability of the mxm matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -- infinite-component systems of hydrodynamic type for which the infinite matrix V(u) is `sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.Comment: 36 pages, the classification results and proofs are refined. A section on generating functions is adde

Freight distribution model predictions compared: a test of hypotheses

Abstract. The transportation problem and a doubly constrained gravity model with a power deterrence function are used to find predictions of a number of 134 x 134 freight matrices detailing tonnages moved in Great Britain in 1972. The matrices detail movements by thirty commodity groups, and predictions are obtained for movements by road for all but one of the commodities and for the principal items carried by rail. These predicted matrices are used to examine a number of questions. The relationships between some alternative goodness-of-fit statistics are examined to establish which commodities are best modelled by each technique and to point out empirically which statistics give unreliable rankings. Various summary measures of the actual matrices are examined to see if it is possible to predict which matrices will be best modelled by each technique. The modelling techniques are compared to indicate which provides the best predictions for each matrix, and some conclusions are offered on the absolute efficiency of the best models.