Recent zbMATH articles in MSC 05B15https://zbmath.org/atom/cc/05B152021-06-15T18:09:00+00:00WerkzeugA complete solution to the existence of normal bimagic squares of even order.https://zbmath.org/1460.050302021-06-15T18:09:00+00:00"Pan, Fengchu"https://zbmath.org/authors/?q=ai:pan.fengchu"Li, Wen"https://zbmath.org/authors/?q=ai:li.wen|li.wen.2|li.wen.1"Chen, Guangzhou"https://zbmath.org/authors/?q=ai:chen.guangzhou"Xin, Bangying"https://zbmath.org/authors/?q=ai:xin.bangyingSummary: Magic square is an ancient and important subject in combinatorial mathematics, and many kinds of magic square are studied and concerned by many scholars, in particular, the existence of normal bimagic squares has been studied for over one hundred years. In this paper, the existence of the normal bimagic squares of even order is investigated, the ideas of general row (column) bimagic rectangles and magic pairs are introduced, which are applied to our construction, then we obtain the spectrum of the normal bimagic squares of even order, in other words, it is shown that there exists a normal bimagic square of order \(2 u\) if and only if \(u \geq 4\).Algebraic techniques for covering arrays and related structures.https://zbmath.org/1460.050292021-06-15T18:09:00+00:00"Garn, Bernhard"https://zbmath.org/authors/?q=ai:garn.bernhard"Simos, Dimitris E."https://zbmath.org/authors/?q=ai:simos.dimitris-eSummary: In this paper, we extend an existing algebraic modelling technique for covering arrays by considering additional properties which are required when these structures are applied in practice in a branch of software testing called combinatorial testing. Corresponding to these properties, we give semantically equivalent systems of multivariate polynomial equations.
For the entire collection see [Zbl 1409.68021].Weighted \(t\)-way sequences.https://zbmath.org/1460.050152021-06-15T18:09:00+00:00"Garn, Bernhard"https://zbmath.org/authors/?q=ai:garn.bernhard"Simos, Dimitris E."https://zbmath.org/authors/?q=ai:simos.dimitris-eSummary: We define the notion of \textit{weighted} \(t\)-\textit{way sequences}, which is built upon sequence covering arrays. The integration of a weight-based modelling formalism together with partitions of positive integers increases the expressiveness of the generated sequences considerably, and makes them applicable as abstract test sequences for real-world sequence testing problems. Applicability of this concept to real-world testing scenarios is investigated.
For the entire collection see [Zbl 1409.68021].Optimal and efficient designs for fMRI experiments via two-level circulant almost orthogonal arrays.https://zbmath.org/1460.621242021-06-15T18:09:00+00:00"Lu, Xiao-Nan"https://zbmath.org/authors/?q=ai:lu.xiaonan"Mishima, Miwako"https://zbmath.org/authors/?q=ai:mishima.miwako"Miyamoto, Nobuko"https://zbmath.org/authors/?q=ai:miyamoto.nobuko"Jimbo, Masakazu"https://zbmath.org/authors/?q=ai:jimbo.masakazuSummary: In this paper, we investigate a class of optimal circulant \(\{ 0 , 1 \}\)-arrays other than the previously known class of optimal designs for fMRI experiments with a single type of stimulus. We suppose throughout the paper that \(n \equiv 2\pmod 4\) and discuss the asymptotic optimality and the D-efficiency of \(k \times n\) circulant almost orthogonal arrays (CAOAs) with 2 levels (presence/absence of the stimulus), strength 2 and bandwidth 1, denoted by CAOA \(( n , k , 2 , 2 , 1 )\). We show that for \(n \equiv 2\pmod 4\) the largest possible value of \(k\) for statistically optimal CAOA \(( n , k , 2 , 2 , 1 )\) cannot exceed \(n \slash 2\). We also clarify that CAOA \(( n , k , 2 , 2 , 1 )\) with high D-efficiency and \(k\) greater than \(n \slash 2\) can be obtained via perfect binary sequences. By applying algebraic constructions for perfect binary sequences and by computer search, lists of such efficient CAOAs and the new class of optimal CAOAs are provided.Uniform semi-Latin squares and their pairwise-variance aberrations.https://zbmath.org/1460.621212021-06-15T18:09:00+00:00"Bailey, R. A."https://zbmath.org/authors/?q=ai:bailey.rosemary-a"Soicher, Leonard H."https://zbmath.org/authors/?q=ai:soicher.leonard-hSummary: For integers \(n > 2\) and \(k > 0\), an \(( n \times n ) \slash k\) \textit{semi-Latin square} is an \(n \times n\) array of \(k\)-subsets (called \textit{blocks}) of an \(n k\)-set (of \textit{treatments}), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is \textit{uniform} if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform \(( n \times n ) \slash k\) semi-Latin square is Schur optimal in the class of all \(( n \times n ) \slash k\) semi-Latin squares, and here we show that when a uniform \(( n \times n ) \slash k\) semi-Latin square exists, the Schur optimal \(( n \times n ) \slash k\) semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform \(( n \times n ) \slash k\) semi-Latin squares with minimum PV aberration when there exist \(n - 1\) mutually orthogonal Latin squares of order \(n\). These do not exist when \(n = 6\), and the smallest uniform semi-Latin squares in this case have size \(( 6 \times 6 ) \slash 10\). We present a complete classification of the uniform \(( 6 \times 6 ) \slash 10\) semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform \(( ( n + 1 ) \times ( n + 1 ) ) \slash ( ( n - 2 ) n )\) semi-Latin square when there exist \(n - 1\) mutually orthogonal Latin squares of order \(n\), and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform \(( 6 \times 6 ) \slash 10\) semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays \(\operatorname{OA} ( 72 , 6 , 6 , 2 )\).