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Publications (10 of 35) Show all publications
Jäger, G., Öhman, L.-D., Markström, K. & Shcherbak, D. (2024). Enumeration of sets of mutually orthogonal latin rectangles. The Electronic Journal of Combinatorics, 31(1), Article ID #P1.53.
Open this publication in new window or tab >>Enumeration of sets of mutually orthogonal latin rectangles
2024 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 31, no 1, article id #P1.53Article in journal (Refereed) Published
Abstract [en]

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR co-isotopic. In the course of doing this, we perform a complete enumeration of sets of t mutually orthogonal k × n Latin rectangles for k ≤ n ≤ 7, for all t < n up to isotopism, and up to paratopism. Additionally, for larger n we enumerate co-isotopic sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are co-isotopic and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property at each step of their construction. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular hypergraphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

Place, publisher, year, edition, pages
Australian National University Press, 2024
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:umu:diva-222588 (URN)10.37236/9049 (DOI)001183448100001 ()2-s2.0-85187699389 (Scopus ID)
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, 2014-4897
Available from: 2024-04-08 Created: 2024-04-08 Last updated: 2024-04-08Bibliographically approved
Lundqvist, S., Stokes, K. & Öhman, L.-D. (2023). Applying the pebble game algorithm to rod configurations. In: EuroCG 2023: Book of abstracts. Paper presented at The 39th European workshop on computational geometry (EuroCG 2023), Barcelona, Spain, March 29-31, 2023. , Article ID 41.
Open this publication in new window or tab >>Applying the pebble game algorithm to rod configurations
2023 (English)In: EuroCG 2023: Book of abstracts, 2023, article id 41Conference paper, Published paper (Refereed)
Abstract [en]

We present results on rigidity of structures of rigid rods connected in joints: rod configurations. The underlying combinatorial structure of a rod configuration is an incidence structure. Our aim is to find simple ways of determining which rod configurations admit non-trivial motions, using the underlying incidence structure.

Rigidity of graphs in the plane is well understood. Indeed, there is a polynomial time algorithm for deciding whether most realisations of a graph are rigid. One of the results presented here equates rigidity of sufficiently generic rod configurations to rigidity of a related graph. As a consequence, itis possible to determine the rigidity of rod configurations using the previously mentioned polynomial time algorithm. We use this to show that all v3-configurations on up to 15 points and all triangle-free v3-configurations on up to 20 points are rigid in regular position, if such a realisation exists. We also conjecture that the smallest v3-configuration that is flexible in regular position is a previously known 283-configuration. 

National Category
Discrete Mathematics Geometry
Identifiers
urn:nbn:se:umu:diva-215548 (URN)
Conference
The 39th European workshop on computational geometry (EuroCG 2023), Barcelona, Spain, March 29-31, 2023
Available from: 2023-10-22 Created: 2023-10-22 Last updated: 2023-10-23Bibliographically approved
Lundqvist, S., Stokes, K. & Öhman, L.-D. (2023). Exploring the rigidity of planar configurations of points and rods. Discrete Applied Mathematics, 336, 68-82
Open this publication in new window or tab >>Exploring the rigidity of planar configurations of points and rods
2023 (English)In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 336, p. 68-82Article in journal (Refereed) Published
Abstract [en]

In this article we explore the rigidity of realizations of incidence geometries consisting of points and rigid rods: rod configurations. We survey previous results on the rigidity of structures that are related to rod configurations, discuss how to find realizations of incidence geometries as rod configurations, and how this relates to the 2-plane matroid. We also derive further sufficient conditions for the minimal rigidity of k-uniform rod configurations and give an example of an infinite family of minimally rigid 3-uniform rod configurations failing the same conditions. Finally, we construct v3-configurations that are flexible in the plane, and show that there are flexible v3-configurations for all sufficiently large values of v.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Combinatorial rigidity, Incidence geometry, Rod configuration
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-208092 (URN)10.1016/j.dam.2023.03.030 (DOI)000983170400001 ()2-s2.0-85153509834 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, 2020.0001Knut and Alice Wallenberg Foundation, 2020.0007
Available from: 2023-05-09 Created: 2023-05-09 Last updated: 2023-09-05Bibliographically approved
Jäger, G., Markström, K., Shcherbak, D. & Öhman, L.-D. (2023). Small youden rectangles, near youden rectangles, and their connections to other row-column designs. Discrete Mathematics & Theoretical Computer Science, 25(1), Article ID 9.
Open this publication in new window or tab >>Small youden rectangles, near youden rectangles, and their connections to other row-column designs
2023 (English)In: Discrete Mathematics & Theoretical Computer Science, ISSN 1462-7264, E-ISSN 1365-8050, Vol. 25, no 1, article id 9Article in journal (Refereed) Published
Abstract [en]

In this paper we first study k × n Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where k = n − 1, in a large scale computer search. In particular, we verify the previous counts for (n, k) = (7, 3), (7, 4), and extend this to the cases (11, 5), (11, 6), (13, 4) and (21, 5). For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call near Youden rectangles. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.

Place, publisher, year, edition, pages
Centre pour la Communication Scientifique Directe (CCSD), 2023
Keywords
block designs, row-column designs, Youden squares
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:umu:diva-206792 (URN)10.46298/DMTCS.6754 (DOI)2-s2.0-85152096973 (Scopus ID)
Funder
Swedish Research Council, 2014-4897Swedish National Infrastructure for Computing (SNIC)eSSENCE - An eScience Collaboration
Available from: 2023-04-24 Created: 2023-04-24 Last updated: 2023-08-18Bibliographically approved
Lundqvist, S., Stokes, K. & Öhman, L.-D. (2023). When is a planar rod configuration infinitesimally rigid?. Discrete & Computational Geometry
Open this publication in new window or tab >>When is a planar rod configuration infinitesimally rigid?
2023 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed) Epub ahead of print
Abstract [en]

We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

Place, publisher, year, edition, pages
Springer Nature, 2023
Keywords
Combinatorial rigidity, Hypergraphs, Incidence geometries, Parallel redrawings, Rod configurations
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:umu:diva-218895 (URN)10.1007/s00454-023-00617-7 (DOI)2-s2.0-85180169240 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, 2020.0001Knut and Alice Wallenberg Foundation, 2020.0007
Available from: 2024-01-04 Created: 2024-01-04 Last updated: 2024-01-04
Öhman, L.-D. (2019). Are Induction and Well-Ordering Equivalent?. The Mathematical intelligencer, 41(3), 33-40
Open this publication in new window or tab >>Are Induction and Well-Ordering Equivalent?
2019 (English)In: The Mathematical intelligencer, ISSN 0343-6993, E-ISSN 1866-7414, Vol. 41, no 3, p. 33-40Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Springer, 2019
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-163654 (URN)10.1007/s00283-019-09898-4 (DOI)000482242600007 ()2-s2.0-85065446295 (Scopus ID)
Available from: 2019-10-31 Created: 2019-10-31 Last updated: 2023-03-24Bibliographically approved
Öhman, L.-D. (2019). Romarna var inte så avancerade.
Open this publication in new window or tab >>Romarna var inte så avancerade
2019 (Swedish)Other (Other (popular science, discussion, etc.))
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-188339 (URN)
Note

Publicerad 2019-09-26.

Texten publicerad i Forskning & framsteg, ISSN:0015-7937, nr 8/2019

Available from: 2021-10-06 Created: 2021-10-06 Last updated: 2021-10-06Bibliographically approved
Jäger, G., Markström, K., Öhman, L.-D. & Shcherbak, D. (2019). Triples of Orthogonal Latin and Youden Rectangles of small order. Journal of combinatorial designs (Print), 27(4), 229-250
Open this publication in new window or tab >>Triples of Orthogonal Latin and Youden Rectangles of small order
2019 (English)In: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 27, no 4, p. 229-250Article in journal (Refereed) Published
Abstract [en]

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal k × n Latin rectangles for k ≤ n ≤ 7. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of k×8 rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of 4 × 8 rectangles, formed by juxtaposing two   selected copies of complete sets of MOLS of order 4.

National Category
Discrete Mathematics
Identifiers
urn:nbn:se:umu:diva-158857 (URN)10.1002/jcd.21642 (DOI)000459040800001 ()2-s2.0-85059030594 (PubMedID)
Available from: 2019-05-13 Created: 2019-05-13 Last updated: 2019-05-23Bibliographically approved
Raman-Sundström, M. & Öhman, L.-D. (2018). Mathematical fit: a case study. Philosophia mathematica, 26(2), 184-210
Open this publication in new window or tab >>Mathematical fit: a case study
2018 (English)In: Philosophia mathematica, ISSN 0031-8019, E-ISSN 1744-6406, Vol. 26, no 2, p. 184-210Article in journal (Refereed) Published
Abstract [en]

Mathematicians routinely pass judgments on mathematical proofs. A proof might be elegant, cumbersome, beautiful, or awkward. Perhaps the highest praise is that a proof is right, that is that the proof fits the theoremin an optimal way. It is also common to judge that a proof fits better than another, or that a proof does not fit a theorem at all. This paper attempts to clarify the notion of mathematical fit. We suggest six criteria that distinguish proofs as being more or less fitting, and provide examples from several different mathematical fields.

Place, publisher, year, edition, pages
Cary: Oxford University Press, 2018
Keywords
Fit, aesthetics, explanation
National Category
Philosophy Mathematics
Research subject
Aesthetics; Mathematics
Identifiers
urn:nbn:se:umu:diva-124162 (URN)10.1093/philmat/nkw015 (DOI)000439703300003 ()2-s2.0-85053033011 (Scopus ID)
Available from: 2016-07-21 Created: 2016-07-21 Last updated: 2018-11-01Bibliographically approved
Öhman, L.-D. (2016). A Beautiful Proof by Induction. Journal of Humanistic Mathematics, 6(1), 73-85
Open this publication in new window or tab >>A Beautiful Proof by Induction
2016 (English)In: Journal of Humanistic Mathematics, E-ISSN 2159-8118, Vol. 6, no 1, p. 73-85Article in journal (Refereed) Published
Abstract [en]

The purpose of this note is to present an example of a proof by induction that in the opinion of the present author has great aesthetic value. The proof in question is Thomassen’s proof that planar graphs are 5-choosable. I give a self-contained presentation of this result and its proof, and a personal account of why I think this proof is beautiful.

A secondary purpose is to more widely publicize this gem, and hopefully make it part of a standard set of examples for examining characteristics of proofs by induction.

National Category
Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-118552 (URN)10.5642/jhummath.201601.06 (DOI)000388610000005 ()
Available from: 2016-03-23 Created: 2016-03-23 Last updated: 2024-03-05Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-7040-4006

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