Appendix: Vector, Tensor, and Matrix Notation

2018 ◽  
Author(s):  
Kevin A. O'Neill
Keyword(s):  
Matrix Notation

Complex Chi-Squares: Matrix Notation and Calculation

1972 ◽  
Vol 30 (3) ◽  
pp. 743-746 ◽  
Author(s):  
Edward F. Gocka
Keyword(s):  
Computational Algorithms ◽  
Behavioral Sciences ◽  
Matrix Notation ◽  
Standard Formula ◽  
The Matrix ◽  
Matrix Formula

A matrix formula available for the calculation of complex chi-squares allows several computational variations, each of which requires fewer steps than the standard formula. However, neither the matrix formula nor the associated computational algorithms have been given adequate exposure in statistical texts for the behavioral sciences. This paper reintroduces the formula, expands the notation, and shows how several computational variations can be derived.

Matrix Notation

1983 ◽  
pp. 5-32
Author(s):  
A. R. G. Heesterman
Keyword(s):  
Matrix Notation

II.—On Fitting Polynomials to Data with Weighted and Correlated Errors

1935 ◽  
Vol 54 ◽  
pp. 12-16 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Positive Definite ◽  
Correlated Errors ◽  
Positive Definite Quadratic Form ◽  
Matrix Notation ◽  
Image Position

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

II.—A Matrix Notation for Mendelian Populations

1934 ◽  
Vol 53 ◽  
pp. 7-25 ◽  
Author(s):  
Lancelot Hogben
Keyword(s):  
Contingency Table ◽  
Single Gene ◽  
Sex Linkage ◽  
Matrix Notation ◽  
Usual Type ◽  
Preliminary Account ◽  
Male Sex

Those who have had experience of problems involving inbreeding or the correlation of relatives in a population with specified proportions of Mendelian genotypes will have realised what difficulties reside in the absence of a convenient notation. In the absence of such a notation difficulties arise more from the unwieldiness of the expressions obtained than the abstruseness of the problems encountered. The familiar chessboard diagrams of Mendelian hybridisation and the form of a contingency table for relatives alike suggest that a matrix notation might be generalised to take into account both sex-linked inheritance and the more usual type of transmission. A determinant form would meet some of the requirements of autosomal transmission; but is not adapted to the asymmetrical case, when the male sex cannot be heterozygous. In extending to more remote relationships a previous inquiry (3) into fraternal and filial correlations involving sex linkage, the writer has found it useful to employ a system of operations applicable to other types of inquiry. The former investigation is not yet complete. In the meantime it seemed possible that the method used might prove suggestive to other workers. A preliminary account of the fundamental operations is here given in so far as they are relevant to single gene substitutions.

Position Analysis of Spatial Mechanisms

1984 ◽  
Vol 106 (2) ◽  
pp. 252-255 ◽  
Author(s):  
J. Llinares ◽  
A. Page
Keyword(s):  
Computational Algorithm ◽  
Matrix Notation ◽  
Position Analysis ◽  
Spatial Mechanisms ◽  
Lower Pair ◽  
Simple Equations

In this paper matrix notation is used to develop a computational algorithm for position analysis of spatial mechanisms having revolute (R) and prismatic (P) pairs. Since all lower pairs are equivalent to some combination of R and P pairs, the method works for spatial mechanisms containing any lower pair. By using the method, spatial mechanisms are describable by simple equations which are easily programmable.

A Discrete Element Method for the Analysis of Plane Elasto-Plastic Stress Problems

1966 ◽  
Vol 17 (1) ◽  
pp. 83-104 ◽  
Author(s):  
G. G. Pope
Keyword(s):  
Discrete Element Method ◽  
Plane Stress ◽  
Discrete Element ◽  
Stress Strain ◽  
Numerical Examples ◽  
Matrix Notation ◽  
The Matrix ◽  
Triangular Elements ◽  
Plastic Stress ◽  
Element Method

SummaryA procedure is developed for the analysis of plane stress problems when yielding occurs locally. The region is divided into triangular elements and the deformation is analysed on a step-by-step basis, using the matrix notation developed by Argyris. The simple expressions which are derived for the element properties are applicable with any stress-strain relations which are stable and time-independent. Simple numerical examples are given.

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