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.     

An analysis of the role of users in the total r&d portfolios of scientific instrument firms

This paper describes a real-time investigation of the total new product portfolios of nine product departments in the scientific instrument industry. It was found that within these portfolios only a few new products represented a major advance in functional performance. Most new products offered only incremental performance improvement, and many ‘new’ products were direct copies of competitors ' offerings.The scientific instrument innovation process is described in seven stages: recognition of need, invention of solution, prototype development, use in application, precommercial information dissemination, product engineering, and commercialization. In those cases where users were involved in the majority of the first five stages of the innovation process, the innovation was either a major functional improvement or a minor improvement that had not been initiated as a direct response to a competitor's product introduction. Direct copies and minor improvements that were competitive responses are notable for the absence of user involvement in the innovation process.Finally, the decision of a manufacturer to commercialize an innovation was unrelated to the level of user activity in the prior stages of the innovation process.     

Polynomial Root Clustering

A sequence of tests on derived polynomials to be strictly Hurwitz polynomials is shown to be equivalent to a given (typically real) polynomial having all its zeros in an open sector, symmetric with respect to the real axis, in the left half-plane. The number of tests needed is at most 1 + ?(ln k)/(ln 3)?, where k is the integer associated with the central angle π/k of the sector. An extension of this result on the sector as a region of root clustering is given which shows that only a limited number of tests are needed to verify that the roots are clustered in a region composed as the intersection of a set of primative (sector-like) regions. The results reported evolve from application of a collection of mappings on the complex plane defined by a particular collection of Schwarz-Christoffel transformations.     

Infant visual acuity as a function of viewing distance.

Dynamic retinoscopy has suggested that near vision may be more acute than far vision during early infancy. To test this, acuity thresholds were determined by presenting square wave gratings in a preference paradigm to 1- and 2-month-old human infants at 4 viewing distances. Gratings were paired with unpatterned fields; direction of first fixation was the dependent measure. Infants exhibited the same acuity at each of the distances at which gratings were presented. The results were interpreted as compatible with the fact that considerable optical defocusing does not seriously affect a visual system, such as the infant's, that is sensitive only to low spatial frequencies.     

Effects of social comparison on aggression and regression in groups of young children.

The influence of negative, equal, and positive social comparison and of nonsocial comparison upon 4- and 5-year-old black children's subsequent aggressive and regressive behavior in 3-member groups was investigated. The group behavior of boys included more physical agression following negative social comparison than the other treatments, and their group behavior also consisted of more nonverbal teasing behavior following the negative comparison treatment than that of the equal and nonsocial comparison groups. When the behavior of the nontarget partners was controlled, children initiated more physical aggression, nonverbal teasing, and regression after experiencing negative social comparison with the partners than after following the other treatments. There was some evidence to support the reciprocal influence of children's aggressive behavior on each other, particularly for boys following imbalanced social comparison treatments.     

Infant approach behavior as related to attachment.

The chief purpose of this study was to identify characteristics that distinguish approach as an attachment behavior from approach serving other behavioral systems. Locomotor approaches of 16 male and 10 female white middle-class infants to an attachment figure (the mother) and to a nonattachment figure (the visitor-observer) were examined under naturalistic conditions at home. Observations were made during 4-hour home visits at 3-week intervals; those from 26 to 54 weeks were used. When approach was examined in a free-choice situation and without regard to behavioral context, infants approached the mother proportionally (though slightly) more often than the visitor (p less than .05). In 2 behavioral contexts, however, spontaneous infant approaches were sharply differential to the mother; approaches accompanied by crying and approaches terminating in a pickup appeal were directed almost exclusively to the mother (p less than .0001 for both). Touching upon completion of the approach was not differential, except in the context of a pickup appeal. Object-oriented approaches were more often directed to the visitor than to the mother (p less than .002) and may be either exploratory or affiliative. It is suggested that neither approach nor touching can be assumed to serve the attachment system without consideration of context- both environmental and behavioral.     

Self-evaluation, social reinforcement, and academic achievement of black and white schoolchildren.

Black and white male schoolchildren covertly evaluated their own performance at a verbal task administered by black and white male experimenters. It was hypothesized that academically successful children would be predisposed to appraise themselves favorably, whereas relatively unsuccessful pupils would be biased toward self-criticism. The covert self-evaluations were assumed to represent at internalization of early experiences of predominantly positive or negative social reinforcement from adult socializing agents. The predicition for self-criticism was upheld in the white experimenter condition but not in the black experimenter condition. Relationships consistent with the theory were found between subjects' self-appraisals and their retrospective reports of positive and negative reinforcements received from parents and teachers in various typical situations. These relationships were more evident for black subjects than for white subjects. The extent to which children's self-praise and self-criticism mediated affect was assessed by means of a color conditioning technique. Contrary to predicition, possibly due to the easy nature of the experimental task.