Recent zbMATH articles in MSC 05B30https://zbmath.org/atom/cc/05B302021-06-15T18:09:00+00:00WerkzeugOn cyclic decompositions of the complete graph into the bipartite generalized Petersen graph \(P ( n , 3 )\).https://zbmath.org/1460.051682021-06-15T18:09:00+00:00"Wannasit, Wannasiri"https://zbmath.org/authors/?q=ai:wannasit.wannasiriSummary: A uniformly-ordered \(\rho \)-labeling (also known as a \(\rho^{+ +}\)-labeling) of a bipartite graph was introduced by \textit{S. I. El-Zanati} et al. [Australas. J. Comb. 24, 209--219 (2001; Zbl 0983.05063)]. Such a labeling of a bipartite graph \(G\) with \(m\) edges yields a cyclic \(G\)-decomposition of \(K_{2 m t + 1}\) for every positive integer \(t\). Here we show that for every even integer \(n \geq 8\), the generalized Petersen graph \(P ( n , 3 )\) admits a \(\rho^{+ +}\)-labeling and hence cyclically decomposes \(K_{6 n t + 1}\) for all positive integers \(t\).Three-class association schemes from partial geometric designs.https://zbmath.org/1460.052032021-06-15T18:09:00+00:00"Qu, Jing"https://zbmath.org/authors/?q=ai:qu.jing"Lei, Jianguo"https://zbmath.org/authors/?q=ai:lei.jianguoSummary: In this paper, we show that three-class association schemes can be constructed from certain partial geometric designs. First, we show that partial geometric designs with exactly three indices: \( r - n\), \(\lambda_1\) and \(\lambda_2\) give rise to three-class association schemes, using the method of \textit{H. Beker} and \textit{W. Haemers} [J. Comb. Theory, Ser. A 28, 64--81 (1980; Zbl 0425.05009)]. Second, we describe parameter sets of certain partial geometric designs that also give rise to three-class association schemes.The effect of points fattening on del Pezzo surfaces.https://zbmath.org/1460.141182021-06-15T18:09:00+00:00"Lampa-Baczyńska, Magdalena"https://zbmath.org/authors/?q=ai:lampa-baczynska.magdalenaIn the paper under review, the author studies the effect of points fattening on del Pezzo surfaces. Let \(S_{r}\) be the del Pezzo surface obtained by blowing up \(r\) general points in the complex projective plane with \(r \in \{1, \dots,8\}\), and denote by \(K_{S_{r}}\) the canonical divisor of \(S_{r}\). In order to formulate the main result of the paper under review, we need to define the initial degree.
Let \(X\) be a smooth projective variety with a fixed ample line bundle \(L\). Let \(Z\) be a reduced subscheme of \(X\) defined by the ideal sheaf \(\mathcal{I}_{Z}\). For a positive integer \(m\), the initial degree with respect to \(L\) of the subscheme \(mZ\) is the following number \[\alpha(mZ) = \alpha\bigg(\mathcal{I}_{Z}^{(m)}\bigg) := \min \bigg\{ d: \, H^{0}(X, dL \otimes \mathcal{I}_{Z}^{(m)}) \neq 0 \bigg\}.\]
The initial sequence, with respect to \(L\), of a subscheme \(Z\) is the sequence of the form \[\alpha(Z), \alpha(2Z), \alpha(3Z), \dots\,.\] It is worth noticing that the initial sequence is weakly growing, i.e., \(\alpha(mZ) \leq (nZ)\) provided that \(n\geq m\), and the sequence is subadditive, i.e., \(\alpha((n+m)Z) \leq \alpha(nZ) + \alpha(mZ)\).
The main goal of the paper under review is to classify finite sets of points in del Pezzo surfaces for which the fattening effect is the smallest possible. More precisely, for each del Pezzo surface \(S_{r}\) the author establishes the maximal integer \(m\) such that the following sequence holds \[\alpha(Z) = \alpha(2Z) = \cdots = \alpha(mZ) = 1,\] where \(Z\) is a finite set of points in \(S_{r}\), and the initial degree is computed with respect to the anticanonical divisor \(-K_{S_{r}}\).
In the second part of the paper, the author shows a Chudnovsky-type statement for del Pezzo surfaces having \(-K_{S_{r}}\) very ample (and the initial degree is computed with respect to this divisor).
Theorem. Let \(1\leq r \leq 6\) and \(Z \subset S_{r}\) be a finite set of points such that \(\alpha = \alpha(Z) \geq 2\), then we have \[\frac{\alpha(mZ)}{m} \geq \frac{\alpha - 1}{2}.\]
Reviewer: Piotr Pokora (Kraków)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.