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Some matters of great balance
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis is based on four papers dealing with two different areas of mathematics.Paper I–III are in combinatorics, while Paper IV is in mathematical physics.In combinatorics, we work with design theory, one of whose applications aredesigning statistical experiments. Specifically, we are interested in symmetric incompleteblock designs (SBIBDs) and triple arrays and also the relationship betweenthese two types of designs.In Paper I, we investigate when a triple array can be balanced for intersectionwhich in the canonical case is equivalent to the inner design of the correspondingsymmetric balanced incomplete block design (SBIBD) being balanced. For this we derivenew existence criteria, and in particular we prove that the residual designof the related SBIBD must be quasi-symmetric, and give necessary and sufficientconditions on the intersection numbers. We also address the question of whenthe inner design is balanced with respect to every block of the SBIBD. We showthat such SBIBDs must possess the quasi-3 property, and we answer the existencequestion for all know classes of these designs.As triple arrays balanced for intersections seem to be very rare, it is natural toask if there are any other families of row-column designs with this property. In PaperII we give necessary and sufficient conditions for balanced grids to be balancedfor intersection and prove that all designs in an infinite family of binary pseudo-Youden designs are balanced for intersection.Existence of triple arrays is an open question. There is one construction of aninfinite, but special family called Paley triple arrays, and one general method forwhich one of the steps is unproved. In Paper III we investigate a third constructionmethod starting from Youden squares. This method was suggested in the literaturea long time ago, but was proven not to work by a counterexample. We show interalia that Youden squares from projective planes can never give a triple array bythis method, but that for every triple array corresponding to a biplane, there is asuitable Youden square for which the method works. Also, we construct the familyof Paley triple arrays by this method.In mathematical physics we work with solitons, which in nature can be seen asself-reinforcing waves acting like particles, and in mathematics as solutions of certainnon-linear differential equations. In Paper IV we study the non-commutativeversion of the two-dimensional Toda lattice for which we construct a family ofsolutions, and derive explicit solution formulas.

##### Place, publisher, year, edition, pages
Sundsvall: Mid Sweden University , 2013. , 60 p.
##### Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 144
##### Keyword [en]
Balanced incomplete block design. Triple array. Balanced grid. Pseudo- Youden design. Youden square. Inner balance. Balanced for intersection. Soliton. Two-dimensional Toda lattice.
Mathematics
##### Identifiers
ISBN: 978-91-87103-67-4OAI: oai:DiVA.org:miun-18757DiVA: diva2:616548
##### Supervisors
Available from: 2013-04-17 Created: 2013-04-17 Last updated: 2013-04-17Bibliographically approved
##### List of papers
1. Inner balance of symmetric designs
Open this publication in new window or tab >>Inner balance of symmetric designs
2014 (English)In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 71, no 2, 247-260 p.Article in journal (Refereed) Published
##### Abstract [en]

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.

Springer, 2014
##### Keyword
Symmetric design, Triple array, Balanced for intersection, Quasi-3 design, Inner design with respect to a block, Quasi-symmetric design
##### National Category
Discrete Mathematics
##### Identifiers
urn:nbn:se:miun:diva-14627 (URN)10.1007/s10623-012-9730-2 (DOI)000332869500004 ()2-s2.0-84897042423 (ScopusID)
##### Projects
Inner balance of designs
##### Note

Published online july 2012

Available from: 2011-10-21 Created: 2011-10-21 Last updated: 2014-04-17Bibliographically approved
2. Pseudo-Youden designs balanced for intersection
Open this publication in new window or tab >>Pseudo-Youden designs balanced for intersection
2011 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, Vol. 141, no 6, 2030-2034 p.Article in journal (Refereed) Published
##### Abstract [en]

If the row-column intersections of a row-column design $\mathcal{A}$ form a balanced incomplete block design, then $\mathcal{A}$ is said to be \emph{balanced for intersection}. This property was originally defined for triple arrays by McSorley et al. (2005a), section 8, where an example was presented and questions of existence were raised and discussed. We give sufficient conditions for the class of balanced grids in order to be balanced for intersection,  and prove that a family of binary pseudo-Youden designs has this property.

##### Keyword
Row-column design, Pseudo-Youden design, Balanced grid, Triple array
##### National Category
Discrete Mathematics
##### Identifiers
urn:nbn:se:miun:diva-12458 (URN)10.1016/j.jspi.2010.12.014 (DOI)000288308900003 ()2-s2.0-79651469655 (ScopusID)
Available from: 2011-01-06 Created: 2010-12-07 Last updated: 2013-04-17Bibliographically approved
3. Triple arrays and Youden squares
Open this publication in new window or tab >>Triple arrays and Youden squares
2015 (English)In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 75, no 3, 429-451 p.Article in journal (Refereed) Published
##### Abstract [en]

This paper addresses the question of when triple arrays can be constructed from Youden squares by removing a column together with the symbols therein, and then exchanging the role of columns and symbols. The scope of the investigation is limited to the standard case of triple arrays with {Mathematical expression}. For triple arrays with {Mathematical expression} it is proven that they can never be constructed in this way, and for triple arrays with {Mathematical expression} it is proven that there always exists a suitable Youden square and a suitable column for this construction. Further, it is proven that Youden square constructed from a certain family of difference sets never give rise to triple arrays in this way but always gives rise to double arrays. Finally, it is proven that all triple arrays in the single known infinite family, the Paley triple arrays, can all be constructed in this way for some suitable choice of Youden square and column.

##### Keyword
Triple array. Youden square. Symmetric incomplete block design.
##### National Category
Discrete Mathematics
##### Identifiers
urn:nbn:se:miun:diva-18549 (URN)10.1007/s10623-014-9926-8 (DOI)000353059700005 ()2-s2.0-84928376222 (ScopusID)
##### Projects
Construction methods for triple arrays Available from: 2013-02-28 Created: 2013-02-28 Last updated: 2015-08-13Bibliographically approved
4. On the noncommutative two-dimensional Toda lattice
Open this publication in new window or tab >>On the noncommutative two-dimensional Toda lattice
##### Keyword
Soliton. Toda lattice.
Mathematics
##### Identifiers
urn:nbn:se:miun:diva-18550 (URN)
Available from: 2013-02-28 Created: 2013-02-28 Last updated: 2013-04-17Bibliographically approved

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Nilson, Tomas
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