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Further results on incomplete (3,2,1)- conjugate orthogonal idempotent Latin squares

F.E. Bennett, Lisheng Wu, L. Zhu
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
An incomplete idempotent Latin square which is orthogonal to its (i, i, k)-con u a e j g t will be called an incomplete (i, j, k)-conjugate orthogonal idempotent Latin square, briefly (i, j, k)-ICOILS.  ...  We shall denote by ICOILS(v, n) an incomplete (3,2,1)-conjugate orthogonal idempotent Latin square of order v based on S and missing a subsquare of order IZ based on X.  ...  For any integer n 2 1, a (1, 3, 2)-ZCOZLS(v, n) exists if v 2 (13/4)n + 88.  ... 
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Incomplete conjugate orthogonal idempotent latin squares

F.E Bennett, L Zhu
<span title="">1987</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let us denote by COILS(v) a (3, 2, 1)-conjugate orthogonal idempotent Latin square of order v, and by ICOILS(v, n) an incomplete COILS(v) missing a sub-COILS(n).  ...  Moreover, for 2 ~< n <~ 6, it is shown that an ICOILS(v; n) exists for all v ~> 3n + 1 with very few possible exceptions.  ...  In [14] , Phelps further proved that a (3, 1, 2) (or (2, 3, 1))-conjugate orthogonal Latin square exists for all orders v ~= 2, 6 and that a (3, 2, 1) (or (1, 3, 2) )-conjugate orthogonal Latin square  ... 
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Latin Squares with Self-Orthogonal Conjugates

Frank E. Bennett, Hantao Zhang
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
on 6 vertices as a conjugate orthogonal Latin square graph.  ...  In this paper, we investigate the existence of idempotent Latin squares for which each conjugate is orthogonal to precisely its own transpose.  ...  A quasigroup (Latin square) is called self-orthogonal if it is orthogonal to its (2; 1; 3)-conjugate.  ... 
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Page 5803 of Mathematical Reviews Vol. , Issue 91K [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
[Zhu, Lie’] (PRC-SOO) Further results on incomplete (3, 2, 1 )-conjugate orthogonal idempotent Latin squares. Discrete Math. 84 (1990), no. 1, 1-14.  ...  A (3, 2, 1)-conjugate orthogonal Latin square of order v, COILS(v), is a Latin square A = (a;;) of order v and with entries {1,2,---,v} such that a;; = i and A is orthogonal to the Latin square B = (b;  ... 
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Incomplete self-orthogonal latin squares ISOLS(6m + 6, 2m) exist for all m

Katherine Heinrich, Lisheng Wu, L. Zhu
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
An incomplete self-orthogonal latin square of order v with an empty subarray of order n, an ISOLS(v, n) can exist only if v 2 3n + 1.  ...  Zhu, Incomplete self-orthogonal latin squares ISOLS(6m + 6, 2m) exist fo all m, Discrete Mathematics 87 (1991) 281-290.  ...  Analogous to the definition of SOLS(v) and ISOLS(v, n) we can define an (i, i, k)-con u a e j g t orthogonal (idempotent) latin square and an incomplete (i, j, k)-conjugate orthogonal (idempotent) latin  ... 
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Author index volume 84 (1990)

<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
On the set coincidence game Rosenfeld, M., The number of cycles in 2-factors of cubic graphs (3) 241-254 (3) 303-307  ...  Zhu, Further results on incomplete (3,2, l)-conjugate orthogonal idempotent Latin squares Bennett, F.E., B. Du and L.  ...  Zhu, On the existence of (v, 7, 1)-perfect Mendelsohn (3) 309-313 Knoblauch, T., Using triangles to partition a disk (Note) (2) 209-211 Lefevre-Percsy, C., and L. van Nypelseer, Finite rank 3 geometries  ... 
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Automated reasoning and exhaustive search: Quasigroup existence problems

J. Slaney, M. Fujita, M. Stickel
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nkrwe4pmozafvnd72yxufztpku" style="color: black;">Computers and Mathematics with Applications</a> </i> &nbsp;
Using different programs has allowed us to cross-check the results, helping reliability.  ...  We find this research interesting from several points of view: first, it brings techniques from the field of automated reasoning to bear on a rather different problem domain from that which motivated their  ...  (3, 2, 1)-conjugate orthogonal.  ... 
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Quasigroup identities and Mendelsohn designs

F. E. Bennett
<span title="1989-04-01">1989</span> <i title="Canadian Mathematical Society"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/m5ds6a3cpng5df7duilz7mzivi" style="color: black;">Canadian Journal of Mathematics - Journal Canadien de Mathematiques</a> </i> &nbsp;
A quasigroup (Latin square) which is orthogonal to its (/,y, &)-conjugate is called (/,/, k)-conjugate orthogonal A (2, 1, 3)-conjugate orthogonal quasigroup (Latin square) is more commonly called self-orthogonal  ...  In particular, the results of Lemmas 5.9, 5.10 and 5.12 provide constructions of (3, 2, 1) (and (3, 1, 2))conjugate orthogonal symmetric Latin squares with equal-sized holes.  ... 
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Diagonally cyclic latin squares

Ian M. Wanless
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
Our primary aim is to survey the multitude of applications of Parker squares and to collect the basic results on them together in a single location.  ...  More generally, we consider squares possessing any cyclic automorphism. Such squares will be named after Parker, in recognition of his seminal contribution to the study of orthogonal latin squares.  ...  To state this result we need to define a unipotent latin square as one in which every symbol on the main diagonal is the same, and an idempotent latin square as one in which the symbols on the main diagonal  ... 
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Latin squares with pairwise orthogonal conjugates

F.E. Bennett
<span title="">1981</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Noting that a Latin square is self-orthogonal if it is orthogonal to its (2, 1,3)-conjugate, Brayton et al.  ...  [2] have shown that Latin squares orthogonal to their (2, 1,3)-conjugates exist for all orders n # 2,3 or 6.  ...  Table 5 .3.  ... 
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Latin squares with pairwise orthogonal conjugates

F.E. Bennett
<span title="">1981</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
nt Latin square of order II with all of its conjugates distinct and pairwise orthogonal. It is known that L* contains all ~ufhciently large integers.  ...  1 integers IS such that there is an idempotent Latin square of crder n with all of its conjugates distinct and pair-wise orthogonal.  ...  It is known that the number of distinct conjugates of a Latin square L is always 1, +, 3 3 or 6.  ... 
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Balanced colourings of strongly regular graphs

R.A. Bailey
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given.  ...  Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge.  ...  If there is an idempotent Latin square of order m, then it gives a colouring f with X f (2B 0 + B 2 )X f = X f (2B 1 + B 3 )X f = 2mJ .  ... 
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A Census of Small Latin Hypercubes

Brendan D. McKay, Ian M. Wanless
<span title="">2008</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
In the process, we prove that no 3-ary loop of order n can have exactly n − 1 identity elements (but no such result holds in dimensions other than 3).  ...  Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.  ...  Incompletable latin cuboids A natural method of building latin hypercubes is to add hyperplanes one at a time, as we did in Section 3.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110413115955/http://cs.anu.edu.au/~Brendan.McKay/papers/hypercubes.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4b/5a/4b5a05da9263f37afd350b6f2b2b05e6c4073a46.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/070693874"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory [article]

Torsten Hertig, Jens Philip Höhmann, Ralf Otte
<span title="2014-06-04">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also show that bicomplex numbers encourage the definition of several different kinds of conjugates. One of these treats the elements of J like the usual conjugate treats complex numbers.  ...  A unitary ring of square matrices is an associative hypercomplex algebra by definition.  ...  So, (2) becomes ze i( p· x−Et) . (3) In conformity with the conventions of relativity theory, especially general relativity, we further use Greek indices if the set of indices includes zero and Latin  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.1014v1">arXiv:1406.1014v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rve5u3wxl5bgxebyyhi5j6rnba">fatcat:rve5u3wxl5bgxebyyhi5j6rnba</a> </span>
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Master index of volumes 81–90

<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Zhu, Further results on incomplete (3,2, 1)-conjugate orthogonal idempotent Latin squares Bennett, F.E., B. Du and L.  ...  Zhu, Incomplete self-orthogonal latin squares ISOLS(6m + 6, 2m) exist for all m Hell, P., see Heinrich, K. Hendry, G.R.T., Extending cycles in graphs Hergert, F.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(91)90131-k">doi:10.1016/0012-365x(91)90131-k</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gbxyeqcrdrcb7eqpruwbpkf2xy">fatcat:gbxyeqcrdrcb7eqpruwbpkf2xy</a> </span>
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