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Signatura of magic and Latin integer squares: isentropic clans and indexing

100%
Cameron I., Rogers A., Loly P.
Discussiones Mathematicae Probability and Statistics
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2013
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tom 33
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nr 1-2
121-149
EN
The 2010 study of the Shannon entropy of order nine Sudoku and Latin square matrices by Newton and DeSalvo [Proc. Roy. Soc. A 2010] is extended to natural magic and Latin squares up to order nine. We demonstrate that decimal and integer measures of the Singular Value sets, here named SV clans, are a powerful way of comparing different integer squares. Several complete sets of magic and Latin squares are included, including the order eight Franklin subset which is of direct relevance to magic square line patterns on chess boards. While early examples suggested that lower rank specimens had lower entropy, sufficient data is presented to show that some full rank cases with low entropy possess a set of singular values separating into a dominant group with the remainder much weaker. An effective rank measure helps understand these issues. We also introduce a new measure for integer squares based on the sum of the fourth powers of the singular values which appears to give a useful method of indexing both Latin and magic squares. This can be used to begin cataloging a "library" of magical squares. Based on a video presentation in celebration of George Styan's 75th at LINSTAT2012 and IWMS-21 on 19 July, 2012 at Będlewo, Poland.
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Control treatments in designs with split units generated by Latin squares

100%
Kuriki S., Mejza I., Ozawa K., Mejza S.
Biometrical Letters
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2014
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tom 51
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nr 2
125-142
EN
This paper deals with two-factor experiments with split units. The whole plot treatments occur in a repeated Latin square, modified Latin square or Youden square, while subplot treatments occur in a block design within the whole plots. The statistical properties of the considered designs are examined. Special attention is paid to the case where one of the treatments is an individual control or an individual standard treatment. In addition, we give a brief overview of work on the design of experiments using the considered designs, as well as possible arrangements of controls in the experiments.
3
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Ranks of permutative matrices

86%
Hu X., Johnson C. R., Davis C. E., Zhang Y.
Special Matrices
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2016
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tom 4
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nr 1
233-246
EN
A new type of matrix, termed permutative, is defined and motivated herein. The focus is upon identifying circumstances under which square permutative matrices are rank deficient. Two distinct ways, along with variants upon them are given. These are a special kind of grouping of rows and a type of partition in which the blocks are again permutative. Other, results are given, along with some questions and conjectures.
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