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This chapter develops a set of two-step identification methods for social interactions models with unknown networks, and discusses how the proposed methods are connected to the identification methods for models with known networks. The first step uses linear regression to identify the reduced forms. The second step decomposes the reduced forms to identify the primitive parameters. The proposed methods use panel data to identify networks. Two cases are considered: the sample exogenous vectors span Rn (long panels), and the sample exogenous vectors span a proper subspace of Rⁿ (short panels). For the short panel case, in order to solve the sample covariance matrices’ non-invertibility problem, this chapter proposes to represent the sample vectors with respect to a basis of a lower-dimensional space so that we have fewer regression coefficients in the first step. This allows us to identify some reduced form submatrices, which provide equations for identifying the primitive parameters.

The aim of this article is to obtain a stable tensegrity structure by using the minimum knowledge of the structure.

Three methods have been formulated based on the eigen value decomposition (EVD) and singular value decomposition theorems. These two theorems are being implemented on the matrices, which are computed from the minimal data of the structure. The required minimum data for the structure is the dimension of the structure, the connectivity matrix of the structure and the initial force density matrix computed from the type of elements. The stability of the structure is analyzed based on the rank deficiency of the force density matrix and equilibrium matrix.

The main purpose of this article is to use the defined methods to find (1) the nodal coordinates of the structure, (2) the final force density values of the structure, (3) single self-stress from multiple self-stresses and (4) the stable structure.

By using the defined approaches, one can understand the difference of each method, which includes, (1) the selection of eigenvalues, (2) the selection of nodal coordinates from the first decomposition theorem, (3) the selection of mechanism mode and force density values further and (4) the solution of single feasible self-stress from multiple self-stresses.

The flexibility matrices for the unassembled structural elements are discussed and a more concise form developed appropriate to the fuselage structure. A general method is given for obtaining self‐equilibrating flange load and shear flow systems in a single‐cell fuselage, the primary redundancies being defined by a simple transformation matrix. The particular matrix based on the orthogonal stress systems for a circular section is especially convenient. A slight extension of the equations leads also to alternative statically equivalent stress systems in the cover.

The purpose of this paper is to describe how the method recently developed for mass‐spring systems and frame structures is modified to include the free vibration of trusses.

Here, two methods are presented for calculating the eigenfrequencies of structures. The first approach is graph theoretical and uses graph symmetry. The graph models are decomposed into submodels and healing processes are employed such that the union of the eigenvalues of the healed submodels contain the eigenvalues of the entire model. The second method has an algebraic nature and uses special canonical forms. The present method is illustrated through three simple examples with odd and even number of bays.

The inter‐relation for the mechanical properties of elements is established using new weighted graphs, enabling easy calculation of the eigenvalues involved. Two methods are presented for calculating the eigenfrequencies of the truss structures.

Symmetry is used for easy calculation of the eigenfrequencies of structures.

This paper describes the use of preset and programme parameters with the matrix interpretive scheme (which was considered in Part I) for the preparation of general purpose programmes for an electronic digital computer. The implications of partitioning the matrices when using the force method of analysis are considered in detail and the above techniques are applied to develop general purpose programmes for the calculations encountered when using this method of structural analysis. The general case, when all matrices are partitioned; the displacement method of structural analysis, and the use of magnetic tape storage, are considered in Part III.

Cross tables are omnipresent in management, academia and popular culture. The Matrix has us, despite all criticism, opposition and desire for a way out. This paper draws on the works of three agents of the matrix. The paper shows that Niklas Luhmann criticised Talcott Parsons’ traditional matrix model of society and proceeded to update systems theory, the latest version of which is coded in the formal language of George Spencer Brown. As Luhmann failed to install his updates to all components of his theory platform, however, regular reoccurrences of Parsonian crosstabs are observed, particularly in the Luhmannian differentiation theory, which results in compatibility issues and produces error messages requesting updates. This paper aims to code the missing update translating the basic matrix structure from Parsonian into Spencer Brownian formal language.

This paper draws on work by Boris Hennig and Louis Kauffman and a yet unpublished manuscript by George Spencer Brown, to demonstrate that the latter introduced his cross as a mark to indicate NOR gates in circuit diagrams. The paper also shows that this NOR gate marker has been taken out of and may be observed to contain the tetralemma, an ancient matrix structure already present in traditional Indian logic. It then proceeds to translate the basic structure of traditional contingency tables into a Spencer Brownian NOR equation and to demonstrate the difference this translation makes in the modelling of social systems.

The translation of cross tables from Parsonian into Spencer Brownian formal language results in the design of a both matrix-shaped and compatible test routine that works as a virtual window for the observation of the actually unobservable medium in which a form is drawn, and can be used for consistency checks of expressions coded in Spencer Brownian formal language.

This paper quotes from and discusses a so far unpublished manuscript finalised by Spencer Brown in April 1961. The basic matrix structure is translated from Parsonian into Spencer Brownian formal language. A Spencer Brownian NOR matrix is coded that may be used to detect errors in expressions coded in Spencer Brownian formal language.

A DSIR Sponsored Research Programme on the Development and Application of the Matrix Force Method and the Digital Computer. The present issue gives a summary of the basic theory of the matrix force method together with some necessary extensions for the fuselage problem. The equilibrium conditions for the idealized structure are then examined in detail and the relevant equations of equilibrium established in matrix form.

The strain energy method for the analysis of pin‐jointed redundant frameworks is expressed in matrix form suitable for solution on electronic computers. This is illustrated by its application to a zero‐length launcher framework having sixty‐three members, nine of which are redundant, using the Ferranti Pegasus computer. It is concluded that a framework must be reasonably complex before the use of this method is justified but that problems of greater complexity than would normally be attempted can readily be solved.

The purpose of this paper is to propose an improved method which can shorten the calculation time and improve the calculation efficiency under the premise of ensuring the calculation accuracy for calculating the response of dynamic systems with periodic time-varying characteristics.

An improved method is proposed based on Runge–Kutta method according to the composition characteristics of the state space matrix and the external load vector formed by the reduction of the dynamic equation of the periodic time-varying system. The recursive scheme of the holistic matrix of the system using the Runge–Kutta method is improved to be the sub-block matrix that is divided into the upper and lower parts to reduce the calculation steps and the occupied computer memory.

The calculation time consumption is reduced to a certain extent about 10–35% by changing the synthesis method of the time-varying matrix of the dynamics system, and the method proposed of paper consumes 43–75% less calculation time in total than the original Runge–Kutta method without affecting the calculation accuracy. When the ode45 command that implements the Runge–Kutta method in the MATLAB software used to solve the system dynamics equation include the time variable which cannot provide its specific analytic function form, so the time variable value corresponding to the solution time needs to be determined by the interpolation method, which causes the calculation efficiency of the ode45 command to be substantially reduced.

The proposed method can be applied to solve dynamic systems with periodic time-varying characteristics, and can consume less calculation time than the original Runge–Kutta method without affecting the calculation accuracy, especially the superiority of the improved method of this paper can be better demonstrated when the degree of freedom of the periodic time-varying dynamics system is greater.

This paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models.

Determine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only.

An alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain.

Reduction of software and hardware requirements for determining steady-states of electromagnetic.

A unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form.

Eliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.