An illustrated introduction to Caïssan squares: the magic of chess

We study various properties of n × n Caïssan magic squares. Following the seminal 1881 article by "Ursus" [Henry James Kesson (b. c. 1844)] in The Queen, we define a magic square to be Caïssan whenever it is pandiagonal and knight-Nasik so that all paths of length n by a chess bishop are magic (pandiagonal, Nasik, CSP1-magic) and by a (regular) chess knight are magic (CSP2-magic). We also study Caïssan beauties, which are pandiagonal and both CSP2- and CSP3-magic; a CSP3-path is by a special knight that leaps over 3 instead of 2 squares. Our paper ends with a bibliography of over 100 items (many with hyperlinks) listed chronologically from the 14th century onwards. We give special attention to items by (or connected with) "Ursus": Henry James Kesson (b. c. 1844), Andrew Hollingworth Frost (1819–1907), Charles Planck (1856–1935), and Pavle Bidev (1912–1988). We have tried to illustrate our findings as much as possible, and whenever feasible, with images of postage stamps or other philatelic items

Inference in Multivariate Normal Populations with Structure. Part 2: Inference When Correlations Have Structure

1 online resource (PDF, 107 pages

Inference in Multivariate Normal Populations with Structure. Part 1: Inference on Variances When Correlations Are Known

1 online resource (PDF, 83 pages

Hua\u27s Matrix Equality and Schur Complements

The purpose of this paper is to revisit Hua\u27s matrix equality (and inequality) through the Schur complement. We present Hua\u27s original proof and two new proofs with some extensions of Hua\u27s matrix equality and inequalities. The new proofs use a result concerning Shur complements and a generalization of Sylvester\u27s law of inertia, each of which is useful in its own right

A Matrix Approach to Nonstationary Chains

1 online resource (PDF, 35 pages

MAGIC MOORE-PENROSE INVERSES AND PHILATELIC MAGIC SQUARES WITH SPECIAL EMPHASIS ON THE DANIELS–ZLOBEC MAGIC SQUARE

We study singular magic matrices in which the numbers in the rows and columns and in the two main diagonals all add up to the same sum. Our interest focuses on such magic matrices for which the Moore– Penrose inverse is also magic. Special attention is given to the “Daniels–Zlobec magic square’’ introduced by the British magician and television performer Paul Daniels (b. 1938) and considered by Zlobec (2001); see also Murray (1989, pp. 30–32). We introduce the concept of a “philatelic magic square” as a square arrangement of images of postage stamps so that the associated nominal values form a magic square. Three philatelic magic squares with stamps especially chosen for Sanjo Zlobec are presented in celebration of his 70th birthday; most helpful in identifying these stamps was an Excel checklist by Männikkö (2009)

and

The matrix partial orderings considered are: (1) the star ordering and (2) the minus ordering or rank subtractivity, both in the set of m X n complex matrices, and (3) the Lowner ordering, in the set of m X m matrices. The problems discussed are: (1) inheriting certain properties under a given ordering, (2) preserving an ordering under some matrix multiplications, (3) relationships between an ordering among direct (or Kronecker) and Hadamard products and the corresponding orderings between the factors involved, (4) orderings between generalized inverses of a given matrix, and (5) preserving or reversing a given ordering under generalized inversions. Several generalizations of results known in the literature and a number of new results are derived

Notes on the Distribution of Quadratic Forms in Singular Normal Variables

1 online resource (PDF, 12 pages

A Philatelic Excursion with Jeff Hunter in Probability and Matrix Theory

We present an excursion with Jeff Hunter, visiting some of his research topics. Specifically, we will present some facts about certain people whose work seems to have influenced Jeff in his scientific career; we illustrate our presentation with postage stamps that have been issued in honour of these people. Our main guide is Hunter’s two-volume book entitled Mathematical Techniques of Applied Probability (Academic Press, 1983)