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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.

Abstract [sv]

Denna avhandling baseras på fyra artiklar som behandlar två olika områden avmatematiken. Artikel I-III ligger inom kombinatoriken medan artikel IV behandlarmatematisk fysik.Inom kombinatoriken arbetar vi med designteori som bland annat har tillämpningardå man ska utforma statistiska experiment.I artikel I undersöker vi när en triple array kan vara snittbalanserad vilket i detkanoniska fallet är ekvivalent med den inre designen till den korresponderandesymmetriska balanserade inkompletta blockdesignen (SBIBD) är balanserad. För dettapresenterar vi nya nödvändiga villkor. Speciellt visar vi att den residuala designentill den korresponderande SBIBDen måste vara kvasi-symmetrisk och ger nödvändigaoch tillräckliga villkor för dess blockskärningstal. Vi adresserar ocksåfrågan om när den inre designen är balanserad med avseende på alla SBIBDensblock. Vi visar att en sådan SBIBD måste ha den egenskap som kallas kvasi-3 ochsvarar på existensfrågan för alla kända klasser av sådana designer.Eftersom snittbalanserade triple arrays verkar vara väldigt sällsynta är detnaturligt att fråga om det finns andra familjer av rad-kolumn designer som hardenna egenskap. I artikel II ger vi nödvändiga och tillräckliga villkor för att enbalanced grid ska vara snittbalanserad och visar att alla designer i en oändlig familjav binära pseudo-Youden squares är snittbalanserade.Existensfrågan för triple arrays är öppen fråga. Det finns en konstruktionsmetodför en oändlig men speciell familj kallad Paley triple arrays och så finns det enallmän metod för vilken ett steg är obevisat. I artikel III undersöker vi en tredjekonstruktionsmetod som utgår från Youden squares. Denna metod föreslogs i litteraturenför länge sedan men blev motbevisad med hjälp av ett motexempel. Vivisar bland annat att Youden squares från projektiva plan aldrig kan ge en triplearray med denna metod, men att det för varje triple array som korresponderartill ett biplan, så finns det en lämplig Youden square för vilken metoden fungerar.Vidare konstruerar vi familjen av Paley triple arrays med denna metod.Inom matematisk fysik arbetar vi med solitoner som man i naturen kan få sesom självförstärkande vågor vilka beter sig som partiklar. Inom matematiken ärde lösningar till vissa ickelinjära differentialekvationer. I artikel IV studerar vi dettvådimensionella Toda-gittret för vilken vi konstruerar en familj av lösningar ochäven explicita lösningsformler.

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.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-18757ISBN: 978-91-87103-67-4 (print)OAI: 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.

Place, publisher, year, edition, pages
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 (Scopus ID)
Projects
Inner balance of designs
Note

Published online july 2012

Available from: 2011-10-21 Created: 2011-10-21 Last updated: 2017-05-04Bibliographically 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 (Scopus ID)
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 (Scopus ID)
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
(English)Manuscript (preprint) (Other academic)
Keyword
Soliton. Toda lattice.
National Category
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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