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The efficiency with which a complex electrical network may be analyzed by a digital computer using topological formulas depends largely upon the efficiency with which the trees and directed trees of the corresponding graph are generated. The problem of efficient generation of trees of a graph has been treated rather extensively in the literature. In this paper it is shown that many of the existing techniques in the literature, seemingly so different in their appearance, are actually the variants of the Wang algebra formulation. Thus, they can all be unified. The unified theory enables one to summarize many of these results systematically and to provide a simple and easy way for their derivations.
In the paper we have shown that many of the existing results described in the literature on the problem of generation of trees of a graph, seemingly so different in their appearance, are actually variants of the Wang algebra formulation. Thus, it enables one to provide a unified theory for their derivation and to summarize most of these results systematically.
The main advantage of the Wang algebra formulation lies in the fact that once a desired set of sets, which can be obtained easily by inspection, is derived from a given graph all the remaining manipulations are algebraic. It is not necessary for one to go back to the original graph to see whether a generated term forms a tree (co-tree) or not. However, the techniques are not very efficient for large graphs because duplicated terms are generated during the process. Apart from slowing down the process of generating trees, this limits the sizes of the graphs that can be analyzed since one must retain the set of all trees in computer memory and check each new tree against the list for possible duplication.
The application of the Wang algebra technique to the generation of directed trees and multi-trees has been considered by Chen (1966). A special case of which is recently given by Shinoda (1968). [ABSTRACT FROM AUTHOR]

This paper presents the algebra of structural numbers, a new approach to the analysis of passive and active circuits. The theory is explained and applied to the passive and active circuits with reference to transistor circuits. The method is computer oriented and offers advantages of economizing computer time and space. The method eliminates lengthy algorithms for evaluation of determinants and the necessity of drawing the flow-graphs. [ABSTRACT FROM AUTHOR]

The intent of this paper is to provide a definition of a sociometric clique in the language of graph theory. This problem is viewed from two perspectives: maintaining fidelity to the intuitive notion of a clique; and providing adequate computational mechanics for large bodies of data. Luce's (1950) concept of an n-clique is used, but further qualifications are added. Two statistics or measures with associated probability distributions are defined for testing the adequacy of a subgraph which qualifies according to the definition. [ABSTRACT FROM AUTHOR]

This paper discusses the application of Bayes's theorem to those cases where the true state of the world is not known with certainty. An algorithm is proposed that relaxes the requirement of Bayes's theorem that the true data state be known with certainty by postulating a true but unobservable elementary event, ω, which gives rise to posterior probabilities which reflect the uncertainty of the data. A derivation is presented for the calculation of Bayesian posterior probabilities which uses as its input these probabilities, rather than the true event, o, which is assumed to be unavailable. Suggestions are made as to the application of this modification of Bayes's theorem to cascaded inference processes. [ABSTRACT FROM AUTHOR]

In regression analysis with heteroscedastic and/or correlated errors, the usual assumption is that the covariance matrix Sigma of the errors is completely specified, except perhaps for a scalar multiplier. This condition is relaxed in this paper by assuming only that Sigma has a certain pattern; for example, that Sigma is diagonal or partitionable into a diagonal matrix of sub-matrices. The method used for estimating Sigma is the standard procedure of equating certain quadratic forms of the observations (in this case, squares and products of residuals from regression) to their expectations, and solving for the unknown variances and covariances. A numerical example illustrates the method. [ABSTRACT FROM AUTHOR]

In the paper "Sources of Productivity Increases in the U.S. Passenger Airline Industry," a log-linear cross-section function was estimated with several independent variables of the "dummy" or "shift" form. In this type of estimating procedure, the dummy variable is interpreted as a shift in the intercept due to some discrete event that is thought to influence the dependent variable. However, the intercept of the estimated linear transformed equation is a multiplicative factor in the equation before transformation. The nontransformed equation, of course, has no intercept. For this reason, the impact of technical change on cost cannot be determined from the equation directly. Using some values for each of the continuous independent variables, use the equation to calculate cost. In this case the mean values of the logarithm were used. If the value of each of the dummy variables is zero, then this calculated cost is interpreted as the expected annual cost of running a propeller driven plane for the sample geometric seat miles, fuel-cost, etc. Replicate this procedure with the addition of a value of unity for the first dummy variable. This is interpreted as the expected annual cost of running a turboprop plane for the sample geometric mean values of the continuous independent variables.