The graph-theoretic field model— II. application of multi-terminal representations to field problems

This paper is a sequel to a paper entitled “The Graph-Theoretic Field Model—I: Modelling and Formulations” (1). Herein, the Theory of Multi-Terminal Representations is applied to the Graph-Theoretic Field Model to provide mathematical models of finite elements. The element models are obtained solely from the algebraic building blocks of the Graph-Theoretic Field Model, without recourse to any functional mathematics. The theory of Multi-Terminal Representations is developed for both linear and non-linear problems. Examples of the application of the theory to one- and two-dimensional field problems are presented from heat conduction and electrostatics.     

Space victualling

Victualling in space differs fundamentally from victualling on land or at sea because there is little chance of foraging or of finding local sources of oxygen and water, and all metabolic products have to be carefully collected even if they will ultimately be jettisoned. The need for a deliberate policy for coping with carbon dioxide, faeces, and urine leads naturally, to schemes for recycling. Entropy is increased when food is metabolised: energy will have to be expended for even partial restoration of the status quo.     

The graph-theoretic field model—I. Modelling and formulations

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure.This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.     

Ladder realizations of multivariable positive real functions

Necessary and sufficient conditions are obtained for the realization of an m-variable positive real function (PRF) as the impedence function of a resistively-terminated ladder network of m lossless two-ports connected in cascade. Each two-port is a single-variable lossless ladder with all of its transmission zeros either at the origin or at finity. Conditions are also obtained when each of the two-ports is a Fujisawa-type lowpass ladder.     

Radial localization of odors by human newborns.

To study sensitivity to radial location of an odor source, 20 human newborns, ranging from 16 to 130 hours of age, were presented with a small amount of ammonium hydroxide. The odor source was placed near the nose slightly to the left or right of midline, with its position randomized over repeated trails. Direction of headturn with respect to the odor location and diffuse motor activity were scored from the videotape recordings of the newborns' behavior. It was found that as a group, the newborns turned away from the odor source more frequently than they turned toward it. The tendency to turn away from the odor was stronger in infants who displayed less motor activity after the response. Newborns also exhibited a right bias in the direction of the head movements. It is concluded that a spatially appropriate avoidance response is present in the neonate and that the newborn is innately sensitive to the radial location of an odor.