Random replicators with high-order interactions

We use tools of the equilibrium statistical mechanics of disordered systems to study analytically the statistical properties of an ecosystem composed of N species interacting via random mutual interactions, as well as via deterministic self-interactions of order p>/=2. We show that the main effect of increasing the order of the interactions among the species is to make the system less competitive, in the sense that the fraction of extinct species is greatly reduced. In addition, we find that for p>2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations.     

Using null strain energy functions in compressible finite elasticity to generate exact solutions

In this paper we characterize those strain energy functions in unconstrained nonlinear elasticity that satisfy the equations of equilibrium identically. The idea is to construct a useful, physically reasonable strain–energy function containing one or more components which are null, in such a way that exact solutions may be obtained from the resulting equilibrium equations. We show that the dilatation is a universal null energy while there may be others that depend on the actual problem. To obtain the null energies for a given problem it is often convenient to formulate the variational problem and look at the Euler–Lagrange equations. Specific examples are used to illustrate some of the potential uses of the method in finding exact solutions for physically meaningful constitutive models.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

The dependence of nuclear magnetic resonance (NMR) image contrast on intrinsic and pulse sequence timing parameters

In Nuclear Magnetic Resonance (NMR) the image pixel value is governed by at least three major intrinsic parameters: the spin density N (H), the spin-lattice relaxation time T1, and the spin-spin relaxation time T2. The extent to which the signal is weighted toward one or several parameters is related to the history of the spin system preceding detection. On the simplifying, though not generally warranted assumption that the spin density does not vary significantly in soft tissues, relative tissue contrast can be predicted quantitatively provided the relaxation times are known. Signal intensities and contrast were computed on the basis of the Bloch equations and experimentally determined relaxation times as a function of pulse timing parameters and the data compared with those in images recorded at 0.5T field strength. Significant deviations from the equal density hypothesis were found for gray and white substance. Notably partial saturation but also spin echo and inversion-recovery images are not in full accordance with predictions made on the basis of relaxation times alone.     

The efficacy of exclusive territory assignments to delivery vehicle drivers

A supporting logic for having a vehicle driver exclusively assigned to serve the same territory on every delivery trip is the deepening of the driver’s knowledge of the territory and the customers therein. This contributes to the driver’s proficiency in serving that territory. However, in situations of randomness in day-to-day customer demands, the choice of exclusive territory assignments entails the sacrifice of sub-optimal route configuration. This study quantifies the extent of that sacrifice in order to depict the cost implications of exclusive territory assignments vis-à-vis tactics that keep pace with day-to-day demand fluctuations by allowing for flexibility in the assignments. The study’s analysis of exclusive territory assignments covers those that involve territory sharing among a team of drivers.     

Inflation and bifurcation of compressible spherical membranes

The inflation and bifurcation of spherical membranes is considered. The membrane material is assumed to be isotropic and hyperelastic but may be arbitrarily compressible. Qualitatively the behaviour of compressible membranes is shown to be the same as that of incompressible membranes but specific forms of strain-energy functions are chosen to illustrate possible quantitative differences.