Recent zbMATH articles in MSC 05Bhttps://zbmath.org/atom/cc/05B2024-02-15T19:53:11.284213ZWerkzeugEnumerating Steiner triple systemshttps://zbmath.org/1526.050082024-02-15T19:53:11.284213Z"Heinlein, Daniel"https://zbmath.org/authors/?q=ai:heinlein.daniel"Östergård, Patric R. J."https://zbmath.org/authors/?q=ai:ostergard.patric-r-jSummary: Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14, 796, 207, 517, 873, 771.
{\copyright} 2023 The Authors. \textit{Journal of Combinatorial Designs} published by Wiley Periodicals LLC.Partitioned difference families and harmonious linear spaceshttps://zbmath.org/1526.050162024-02-15T19:53:11.284213Z"Buratti, Marco"https://zbmath.org/authors/?q=ai:buratti.marco"Jungnickel, Dieter"https://zbmath.org/authors/?q=ai:jungnickel.dieterSummary: A linear space is harmonious if it is resolvable and admits an automorphism group acting sharply transitively on the points and transitively on the parallel classes. Generalizing old results by \textit{M. Buratti} et al. [Electron. J. Comb. 17, No. 1, Research Paper R139, 23 p. (2010; Zbl 1204.05027)] we present some difference methods to construct harmonious linear spaces. In particular, it is shown that, for any finite non-singleton subset \(K\) of \(\mathbb{Z}^+\), there are infinitely many values of \(v\) for which there exists a partitioned difference family that is the base parallel class of a harmonious linear space with \(v\) points whose block sizes are precisely the elements of \(K\).A quasidouble of the affine plane of order 4 and the solution of a problem on additive designshttps://zbmath.org/1526.050172024-02-15T19:53:11.284213Z"Pavone, Marco"https://zbmath.org/authors/?q=ai:pavone.marco.1|pavone.marcoSummary: A \(2\)-\((v, k, \lambda)\) block design \((\mathcal{P}, \mathcal{B})\) is additive if, up to isomorphism, \(\mathcal{P}\) can be represented as a subset of a commutative group \((G, +)\) in such a way that the \(k\) elements of each block in \(\mathcal{B}\) sum up to zero in \(G\). If, for some suitable \(G\), the embedding of \(\mathcal{P}\) in \(G\) is also such that, conversely, any zero-sum \(k\)-subset of \(\mathcal{P}\) is a block in \(\mathcal{B}\), then \((\mathcal{P}, \mathcal{B})\) is said to be strongly additive.
In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable \(2\)-\((16, 4, 2)\) design \((\mathbb{F}_4 \times \mathbb{F}_4, \mathcal{B}_2)\), which decomposes into two disjoint isomorphic copies of the affine plane of order four.
An essential part of our construction is a (cyclic) decomposition of the point-plane design of \(\operatorname{AG}(4, 2)\) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem.Totally symmetric quasigroups of order 16https://zbmath.org/1526.050182024-02-15T19:53:11.284213Z"Ginsberg, Hy"https://zbmath.org/authors/?q=ai:ginsberg.hySummary: We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.
{\copyright} 2023 Wiley Periodicals LLC.Magic partially filled arrays on abelian groupshttps://zbmath.org/1526.050192024-02-15T19:53:11.284213Z"Morini, Fiorenza"https://zbmath.org/authors/?q=ai:morini.fiorenza"Pellegrini, Marco Antonio"https://zbmath.org/authors/?q=ai:pellegrini.marco-antonioSummary: In this paper we introduce a special class of partially filled arrays. A magic partially filled array \(\mathrm{MPF}_{\Omega} (m, n; s, k)\) on a subset \(\Omega\) of an abelian group \((\Gamma, +)\) is a partially filled array of size \(m \times n\) with entries in \(\Omega\) such that (i) every \(\omega \in \Omega\) appears once in the array; (ii) each row contains \(s\) filled cells and each column contains \(k\) filled cells; (iii) there exist (not necessarily distinct) elements \(x, y \in \Gamma\) such that the sum of the elements in each row is \(x\) and the sum of the elements in each column is \(y\). In particular, if \(x=y={0}_{\Gamma}\), we have a zero-sum magic partially filled array \({}^0\mathrm{MPF}_{\Omega}(m, n; s, k)\). Examples of these objects are magic rectangles, \(\Gamma\)-magic rectangles, signed magic arrays, (integer or noninteger) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, that is, of an \(\mathrm{MPF}_{\Omega} (m, n; s, k)\) where \(\Omega=\{1, 2, \dots, nk\} \subset \mathbb{Z}\). We also construct zero-sum magic partially filled arrays when \(\Omega\) is the abelian group \(\Gamma\) or the set of its nonzero elements.
{{\copyright} 2023 The Authors. \textit{Journal of Combinatorial Designs} published by Wiley Periodicals LLC.}Existence of small ordered orthogonal arrayshttps://zbmath.org/1526.050202024-02-15T19:53:11.284213Z"Schmidt, Kai-Uwe"https://zbmath.org/authors/?q=ai:schmidt.kai-uwe"Weiß, Charlene"https://zbmath.org/authors/?q=ai:weiss.charleneSummary: We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to \textit{G. Kuperberg} et al. [Geom. Funct. Anal. 27, No. 4, 919--972 (2017; Zbl 1369.05024)].
{{\copyright} 2023 The Authors. \textit{Journal of Combinatorial Designs} published by Wiley Periodicals LLC.}Complex Hadamard graphs and Butson matriceshttps://zbmath.org/1526.050212024-02-15T19:53:11.284213Z"Jacob Chathely, Briji"https://zbmath.org/authors/?q=ai:jacob-chathely.briji"Deore, Rajendra P."https://zbmath.org/authors/?q=ai:deore.rajendra-pSummary: This article introduces complex Hadamard graphs and studies their properties. Using the complete subgraphs of these complex Hadamard graphs, complex Hadamard matrices of order \(n\) are generated, where \(n\) is a multiple of four. An algorithm to construct strongly regular graphs with \(n^2\) vertices using complex Hadamard matrices of order \(n\) is also discussed in this article. MAGMA codes to construct the adjacency matrices of these strongly regular graphs are also developed.Classifications and constructions of minimum size linear sets on the projective linehttps://zbmath.org/1526.050222024-02-15T19:53:11.284213Z"Napolitano, Vito"https://zbmath.org/authors/?q=ai:napolitano.vito"Polverino, Olga"https://zbmath.org/authors/?q=ai:polverino.olga"Santonastaso, Paolo"https://zbmath.org/authors/?q=ai:santonastaso.paolo"Zullo, Ferdinando"https://zbmath.org/authors/?q=ai:zullo.ferdinandoSummary: This paper aims to study linear sets of minimum size on the projective line, that is \(\mathbb{F}_q\)-linear sets of rank \(k\) in \(\mathrm{PG}(1, q^n)\) admitting one point of weight one and having size \(q^{k-1}+1\). Examples of these linear sets have been found by \textit{G. Lunardon} and \textit{O. Polverino} [J. Comb. Theory, Ser. A 90, No. 1, 148--158 (2000; Zbl 0964.51009)] and, more recently, by \textit{D. Jena} and \textit{G. Van de Voorde} [Discrete Math. 344, No. 3, Article ID 112230, 13 p. (2021; Zbl 1456.51006)]. However, classification results for minimum size linear sets are known only for \(k\leq 5\). In this paper we provide classification results for minimum size linear sets admitting two points with complementary weights. We construct new examples (not necessarily with complementary weights) and also study the related \(\Gamma \mathrm{L}(2, q^n)\)-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde [loc. cit.]. The main tool relies on two results by \textit{C. Bachoc} et al. [Math. Proc. Camb. Philos. Soc. 163, No. 3, 423--452 (2017; Zbl 1405.11134); Combinatorica 38, No. 4, 759--777 (2018; Zbl 1413.11115)] on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.Linear and circular single-change covering designs revisitedhttps://zbmath.org/1526.050232024-02-15T19:53:11.284213Z"Chafee, Amanda"https://zbmath.org/authors/?q=ai:chafee.amanda"Stevens, Brett"https://zbmath.org/authors/?q=ai:stevens.brettSummary: A single-change covering design (SCCD) is a \(v\)-set \(X\) and an ordered list \(\mathcal{L}\) of \(b\) blocks of size \(k\) where every pair from \(X\) must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given \(v\), \(k\). In this paper, we use a new recursive construction to solve the existence of circular SCCD\((v, 4, b)\) for all \(v\) and three residue classes of circular SCCD\((v, 5, b)\) modulo 16. We solve the existence of three residue classes of SCCD\((v, 5, b)\) modulo 16. We prove the existence of circular SCCD\((2c(k-1) + 1, k, {c}^2 (2k-2) + c)\), for all \(c \geq 1\), \(k \geq 2\), using difference methods.
{{\copyright} 2023 The Authors. \textit{Journal of Combinatorial Designs} published by Wiley Periodicals LLC.}A generalization of group divisible \(t\)-designshttps://zbmath.org/1526.050242024-02-15T19:53:11.284213Z"Liu, Sijia"https://zbmath.org/authors/?q=ai:liu.sijia"Han, Yue"https://zbmath.org/authors/?q=ai:han.yue"Ma, Lijun"https://zbmath.org/authors/?q=ai:ma.lijun"Wang, Lidong"https://zbmath.org/authors/?q=ai:wang.lidong.1"Tian, Zihong"https://zbmath.org/authors/?q=ai:tian.zihongSummary: \textit{P. J. Cameron} [Discrete Math. 309, No. 14, 4835--4842 (2009; Zbl 1186.05024)] defined the concept of generalized \(t\)-designs, which generalized \(t\)-designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized \(t\)-designs and group divisible \(t\)-designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible \(t\)-designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible \(t\)-designs and almost completely determine their existence for \(t=2, 3\) and small block sizes.
{\copyright} 2023 Wiley Periodicals LLC.Towards the Ryser-Woodall \(\lambda\)-design conjecturehttps://zbmath.org/1526.050252024-02-15T19:53:11.284213Z"Singhi, Navin M."https://zbmath.org/authors/?q=ai:singhi.navin-madhavprasad"Shrikhande, Mohan S."https://zbmath.org/authors/?q=ai:shrikhande.mohan-s"Pawale, Rajendra M."https://zbmath.org/authors/?q=ai:pawale.rajendra-mSummary: Let \(r_1\) and \(r_2\), \((r_1 > r_2)\) be the two replication numbers of a \(\lambda\)-design \(D\). We denote the block size \(|B_j|\) by \(k_j\) and by \(k_j^{^{\prime}}\) (respectively, \(k_j^\ast\)) the number of points with replication number \(r_1\) (respectively, \(r_2\)) in block \(B_j\) of \(D\). Take \(g=\gcd\Big(\frac{r_1-r_2}{\gcd(r_1-1, r_2-1)}, \lambda \Big)\), \(\lambda ={\lambda}_1g\), \(\frac{r_1-r_2}{\gcd(r_1-1, r_2-1)}=gm\), for positive integers \(\lambda_1, m\) and let \(g_1\) be the largest divisor of \(\lambda_1\) such that if \(p\) is a prime dividing \(g_1\), then \(p\) divides \(g\). We obtain the following results:
\begin{itemize}
\item[1.] If \(D\) is a type-1 \(\lambda\)-design then \(k_j^{\prime}\) and \(k_j^\ast\) are divisible by \(gg_1\).
\item[2.] If \(k_j^{\prime}-k_j^\ast\) is divisible by \(gg_1\) for all blocks \(B_j\) of a design \(D\), then \(D\) is a type-1 \(\lambda\)-design.
\end{itemize}
{{\copyright} 2023 Wiley Periodicals LLC.}On the existence of \(k\)-cycle semiframes for even \(k\)https://zbmath.org/1526.050262024-02-15T19:53:11.284213Z"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.3|wang.li.2|wang.li.60|wang.li.70|wang.li|wang.li.25|wang.li.29|wang.li.9|wang.li.61|wang.li.7|wang.li.17|wang.li.11|wang.li.14|wang.li.6|wang.li.16|wang.li.23|wang.li.52|wang.li.13|wang.li.15|wang.li.5|wang.li.20|wang.li.4|wang.li.10|wang.li.24|wang.li.12|wang.li.1|wang.li.62"Ji, Haibo"https://zbmath.org/authors/?q=ai:ji.haibo"Cao, Haitao"https://zbmath.org/authors/?q=ai:cao.haitao.1|cao.haitaoSummary: A \(C_k\)-semiframe of type \(g^u\) is a \(C_k\)-group divisible design of type \(g^u (\mathcal{X}, \mathcal{G}, \mathcal{B})\) in which \(\mathcal{X}\) is the vertex set, \(\mathcal{G}\) is the group set, and the set \(\mathcal{B}\) of \(k\)-cycles can be written as a disjoint union \(\mathcal{B}=\mathcal{P} \cup \mathcal{Q}\) where \(\mathcal{P}\) is partitioned into parallel classes on \(\mathcal{X}\) and \(\mathcal{Q}\) is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition \(\mathcal{X}\) or \(\mathcal{X} \setminus G_j\) for some \(G_j \in \mathcal{G}\). In this paper, we almost completely solve the existence of a \(C_{4k}\)-semiframe of type \(g^u\) for all \(k \geq 1\) and a \(C_{4k+2}\)-semiframe of type \(g^u\) for all \(k \geq 1\) and \(g \equiv 0 \pmod{8k+4}\).
{\copyright} 2023 Wiley Periodicals LLC.The Merino-Welsh conjecture for split matroidshttps://zbmath.org/1526.050272024-02-15T19:53:11.284213Z"Ferroni, Luis"https://zbmath.org/authors/?q=ai:ferroni.luis"Schröter, Benjamin"https://zbmath.org/authors/?q=ai:schroter.benjaminSummary: \textit{C. Merino} and \textit{D. J. A. Welsh} [ibid. 3, No. 2--4, 417--429 (1999; Zbl 0936.05043)] conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.Ehrhart theory of paving and panhandle matroidshttps://zbmath.org/1526.050282024-02-15T19:53:11.284213Z"Hanely, Derek"https://zbmath.org/authors/?q=ai:hanely.derek"Martin, Jeremy L."https://zbmath.org/authors/?q=ai:martin.jeremy-l"McGinnis, Daniel"https://zbmath.org/authors/?q=ai:mcginnis.daniel"Miyata, Dane"https://zbmath.org/authors/?q=ai:miyata.dane"Nasr, George D."https://zbmath.org/authors/?q=ai:nasr.george-d"Vindas-Meléndez, Andrés R."https://zbmath.org/authors/?q=ai:vindas-melendez.andres-r"Yin, Mei"https://zbmath.org/authors/?q=ai:yin.meiSummary: We show that the base polytope \(P_M\) of any paving matroid \(M\) can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of \(P_M\), starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes \textit{L. Ferroni}'s work on sparse paving matroids [Discrete Comput. Geom. 68, No. 1, 255--273 (2022; Zbl 1490.05026)]. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by \textit{L. Ferroni} [``Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids'', Preprint, \url{arXiv:2110.08869}], which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.Detecting arrays for effects of single factorshttps://zbmath.org/1526.050292024-02-15T19:53:11.284213Z"Colbourn, Charles J."https://zbmath.org/authors/?q=ai:colbourn.charles-j"Syrotiuk, Violet R."https://zbmath.org/authors/?q=ai:syrotiuk.violet-rSummary: Determining correctness and performance for complex engineered systems necessitates testing the system to determine how its behavior is impacted by factors and interactions among them. Of particular concern is to determine which settings of single factors (main effects) impact the behavior significantly. Detecting arrays for main effects are test suites that ensure that the impact of each main effect is witnessed even in the presence of \(d\) or fewer other significant main effects. Separation in detecting arrays dictates the presence of at least a specified number of such witnesses. A new parameter, corroboration, enables the fusion of levels while maintaining the presence of witnesses. Detecting arrays for main effects, having various values for the separation and corroboration, are constructed using error-correcting codes and separating hash families. The techniques are shown to yield explicit constructions with few tests for large numbers of factors.
For the entire collection see [Zbl 1519.00033].On perfect sequence covering arrayshttps://zbmath.org/1526.050302024-02-15T19:53:11.284213Z"Gentle, Aidan R."https://zbmath.org/authors/?q=ai:gentle.aidan-r"Wanless, Ian M."https://zbmath.org/authors/?q=ai:wanless.ian-mSummary: A PSCA\((v, t, \lambda)\) is a multiset of permutations of the \(v\)-element alphabet \(\{0, \ldots, v-1\}\), such that every sequence of \(t\) distinct elements of the alphabet appears in the specified order in exactly \(\lambda\) of the permutations. For \(v \geqslant t \geqslant 2\), we define \(g(v, t)\) to be the smallest positive integer \(\lambda\), such that a PSCA\((v, t, \lambda)\) exists. We show that \(g(6, 3) = g(7, 3) = g(7, 4) = 2\) and \(g(8, 3) = 3\). Using suitable permutation representations of groups, we make improvements to the upper bounds on \(g(v, t)\) for many values of \(v \leqslant 32\) and \(3\leqslant t\leqslant 6\). We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.Augmented Aztec bipyramid and dicube tilingshttps://zbmath.org/1526.050312024-02-15T19:53:11.284213Z"Choi, Sunmook"https://zbmath.org/authors/?q=ai:choi.sunmook"Lee, Sangyop"https://zbmath.org/authors/?q=ai:lee.sangyop"Oh, Seungsang"https://zbmath.org/authors/?q=ai:oh.seungsangA polycube is a connected solid figure in \(\mathbb R^3\) formed by combining unit cubes facet to facet. A dicube consists of two unit cubes attached along a facet. The authors consider the enumeration of dicube tilings, where each tiling represents a \(3\)-dimensional tessellation of a polycube using dicubes. In previous related research, the enumeration of domino tilings of polycubes, such as the Aztec diamond and the augmented Aztec diamond, was studied extensively.
Here the authors focus on a three-dimensional analogue, the augmented Aztec bipyramid. This polycube consists of unit cubes and resembles a Platonic octahedron. In this paper, the authors discover a bijection between dicube tilings of the augmented Aztec bipyramid and \(3\)-dimensional Delannoy paths and use this correspondence to determine the number of dicube tilings of the augmented Aztec bipyramid.
The main result of this paper is the following theorem.
Theorem. The number of dicube tilings of the augmented Aztec bipyramid \(P_n\) of order \(n\) is given by \[ \sum_{k=0}^n \binom{n+k}{n-k}\binom{2k}{k}^2. \]
Reviewer: Sinai Robins (São Paulo)From matrix pivots to graphs in surfaces: exploring combinatorics through partial dualshttps://zbmath.org/1526.050382024-02-15T19:53:11.284213Z"Moffatt, Iain"https://zbmath.org/authors/?q=ai:moffatt.iainSummary: To what extent is a graph determined by the trees in it? What changes if we ask this question not for graphs in the abstract, but graphs that are embedded on surfaces? By considering these questions we will see how a collection of seemingly disjoint topics in mathematics are brought together through the idea of a partial dual.
For the entire collection see [Zbl 1519.00033].The rank of the sandpile group of random directed bipartite graphshttps://zbmath.org/1526.050632024-02-15T19:53:11.284213Z"Bhargava, Atal"https://zbmath.org/authors/?q=ai:bhargava.atal"DePascale, Jack"https://zbmath.org/authors/?q=ai:depascale.jack"Koenig, Jake"https://zbmath.org/authors/?q=ai:koenig.jakeSummary: We identify the asymptotic distribution of \(p\)-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös-Rényi random directed graph model, suggesting that the Sylow \(p\)-subgroups of this model may also be Cohen-Lenstra distributed. Our work builds on the results of \textit{S. Koplewitz} [Ann. Comb. 27, No. 1, 1--18 (2023; Zbl 1518.05171)] who studied \(p\)-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of \(p\)-rank differs from the Cohen-Lenstra distribution. Koplewitz [loc. cit.] conjectured that for random balanced bipartite graphs, the expected value of \(p\)-rank is \(O(1)\) for any \(p\). This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.Weighted Tutte-Grothendieck polynomials of graphshttps://zbmath.org/1526.050702024-02-15T19:53:11.284213Z"Chakraborty, Himadri Shekhar"https://zbmath.org/authors/?q=ai:chakraborty.himadri-shekhar"Miezaki, Tsuyoshi"https://zbmath.org/authors/?q=ai:miezaki.tsuyoshi"Zheng, Chong"https://zbmath.org/authors/?q=ai:zheng.chongSummary: In this paper, we introduce the notion of weighted chromatic polynomials of a graph associated to a weight function \(f\) of a certain degree, and discuss some of its properties. As a generalization of this concept, we define the weighted Tutte-Grothendieck polynomials of graphs. When \(f\) is harmonic, we notice that there is a correspondence between the weighted Tutte-Grothendieck polynomials of graphs and the weighted Tutte polynomials of matroids. Moreover, we present some constructions of the weighted Tutte-Grothendieck invariants for graphs as well as the weighted Tutte invariants for matroids. Finally, we give a remark on the categorification of the weighted chromatic polynomials of graphs and weighted Tutte polynomials of matroids.Covering schemes of strength \(t\)https://zbmath.org/1526.051132024-02-15T19:53:11.284213Z"Castoldi, André Guerino"https://zbmath.org/authors/?q=ai:castoldi.andre-guerino"Martinhão, Anderson Novaes"https://zbmath.org/authors/?q=ai:martinhao.anderson-novaes"Monte Carmelo, Emerson L."https://zbmath.org/authors/?q=ai:monte-carmelo.emerson-l"dos Santos, Otávio J. N. T. N."https://zbmath.org/authors/?q=ai:santos.otavio-j-n-t-n-dosSummary: This work brings together several types of combinatorial designs: difference matrices, difference covering arrays and difference schemes by defining the concept of covering scheme of strength \(t\) over an abelian additive group. Connections of covering schemes with orthogonal arrays and covering arrays are also established. We show general results of covering schemes of strength \(t\) using a method based on the factorization of a group and some refinements for particular classes. We apply the previous results to investigate covering schemes having three, four and five factors. Finally, a reformulation of covering schemes in terms of graph theory is established.Switching checkerboards in \((0,1)\)-matriceshttps://zbmath.org/1526.051302024-02-15T19:53:11.284213Z"Ellison, David"https://zbmath.org/authors/?q=ai:ellison.david-j"Jouve, Bertrand"https://zbmath.org/authors/?q=ai:jouve.bertrand"Stone, Lewi"https://zbmath.org/authors/?q=ai:stone.lewiSummary: In order to study \(\mathcal{M}(R, C)\), the set of binary matrices with fixed row and column sums \(R\) and \(C\), we consider submatrices of the form \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), called positive and negative checkerboard respectively. We define an oriented graph of matrices \(G(R, C)\) with vertex set \(\mathcal{M}(R, C)\) and an arc from \(\mathcal{A}\) to \(\mathcal{A}^{\prime}\) indicates you can reach \(\mathcal{A}^{\prime}\) by switching a negative checkerboard in \(\mathcal{A}\) to positive. We show that \(G(R, C)\) is a directed acyclic graph and identify classes of matrices which constitute unique sinks and sources of \(G(R, C)\). Given \(\mathcal{A}, \mathcal{A}^{\prime}\in \mathcal{M}(R, C)\), we give necessary conditions and sufficient conditions on \(\mathcal{M} = \mathcal{A}^{\prime} - \mathcal{A}\) for the existence of a directed path from \(\mathcal{A}\) to \(\mathcal{A}^{\prime}\).
We then consider the special case of \(\mathcal{M}(\mathcal{D})\), the set of adjacency matrices of graphs with fixed degree distribution \(\mathcal{D}\). We define \(G(\mathcal{D})\) accordingly by switching negative checkerboards in symmetric pairs. We show that \(Z_2\), an approximation of the spectral radius \(\lambda_1\) based on the second Zagreb index, is non-decreasing along arcs of \(G(\mathcal{D})\). Also, \(\lambda_1\) reaches its maximum in \(\mathcal{M}(\mathcal{D})\) at a sink of \(G(\mathcal{D})\). We provide simulation results showing that applying successive positive switches to an Erdős-Rényi graph can significantly increase \(\lambda_1\).On multiplicative energy of subsets of varietieshttps://zbmath.org/1526.110052024-02-15T19:53:11.284213Z"Shkredov, Ilya D."https://zbmath.org/authors/?q=ai:shkredov.ilya-dSummary: We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.Subgroups of classical groups that are transitive on subspaceshttps://zbmath.org/1526.200742024-02-15T19:53:11.284213Z"Giudici, Michael"https://zbmath.org/authors/?q=ai:giudici.michael"Glasby, S. P."https://zbmath.org/authors/?q=ai:glasby.stephen-peter"Praeger, Cheryl E."https://zbmath.org/authors/?q=ai:praeger.cheryl-eIn the paper under review, the authors, for each finite classical group \(G\), classify the subgroups of \(G\) which act transitively on a \(G\)-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. The proof uses the classification of the maximal factorisations of almost simple groups (see [\textit{M. Liebeck} et al., Mem. Am. Math. Soc. 432, 151 p. (1990; Zbl 0703.20021)]). As a first application of these results, the authors classify all point-transitive subgroups of automorphisms of finite thick classical generalised quadrangles.
Reviewer: Egle Bettio (Venezia)Covering by planks and avoiding zeros of polynomialshttps://zbmath.org/1526.460072024-02-15T19:53:11.284213Z"Glazyrin, Alexey"https://zbmath.org/authors/?q=ai:glazyrin.alexey"Karasev, Roman"https://zbmath.org/authors/?q=ai:karasev.roman-n"Polyanskii, Alexandr"https://zbmath.org/authors/?q=ai:polyanskii.aleksandr-aAnalysing the proofs in the papers [\textit{Y.-F. Zhao}, Am. Math. Mon. 129, No. 7, 678--680 (2022; Zbl 1494.52018); \textit{O. Ortega-Moreno}, ``The complex plank problem, revisited'', Disc. Comput. Geom. (2022; \url{doi:10.1007/s00454-022-00423-7})], the authors discover the following very general results on zero sets of multivariate polynomials.
Theorem 1. If a polynomial \(P \in \mathbb R [x_1, \dots , x_d]\) of degree \(n\) has a nonzero restriction to the unit sphere \(S^{d-1} \subset \mathbb R^d\) and attains its maximal absolute value on \(S^{d-1}\) at a point \(p\), then \(p\) is at angular distance at least \(\frac{\pi}{2n}\) from the intersection of the zero set of \(P\) with \(S^{d-1}\).
Theorem 2. If a homogeneous polynomial \(P \in \mathbb C[x_1, \dots , x_d]\) of degree \(n\) is not identically zero and attains its maximal absolute value on the unit sphere \(S^{2d-1}\subset \mathbb C^d\) at a point \(p\), then \(p\) is at angular distance at least \(\arcsin\frac{1}{\sqrt{n}}\) from the intersection of the zero set of \(P\) with \(S^{2d-1}\).
They also prove the following result about the zero set inside the unit ball \(B^d \subset \mathbb R^d\): for every nonzero polynomial \(P \in \mathbb R [x_1, \dots , x_d]\) of degree \(n\), there exists a point of \(B^d\) at distance at least \(\frac{1}{n}\) from the zero set of the polynomial~\(P\).
Substituting in the last result a polynomial of the form
\[
P=\prod_{k=1}^n\left(b_k -\sum_{j=1}^d a_{k,j}x_j \right),
\]
one obtains the famous Bang plank covering theorem [\textit{Th. Bang}, Proc. Am. Math. Soc. 2, 990--993 (1951; Zbl 0044.37802)] for the case of planks of equal widths. The authors state the following challenging conjecture which, if true, could be considered as a polynomial generalization of the Bang theorem for arbitrary widths of planks.
Conjecture. Assume that \(P_1, \ldots, P_N \in \mathbb R [x_1, \dots , x_d]\) are nonzero polynomials and \(\delta_1, \ldots, \delta_N > 0\) are such that
\[
\sum_{k=1}^N \delta_k \deg P_k \le 1.
\]
Then there exists a point \(p \in B^d\) such that, for every \(k = 1, \dots ,N\), the point \(p\) is at distance at least \(\delta_k\) from the zero set of \(P_k\).
Reviewer: Vladimir Kadets (Holon)Filling space with hypercubes of two sizes -- the Pythagorean tiling in higher dimensionshttps://zbmath.org/1526.520152024-02-15T19:53:11.284213Z"Führer, Jakob"https://zbmath.org/authors/?q=ai:fuhrer.jakobA tiling of the \(n\)-dimensional Euclidean space into hypercubes is called unilateral if no two hypercubes of the same size share a full facet. A tiling of \(\mathbb R^n\) into hypercubes is called equitransitive if any two hypercubes of the same size can be mapped to each other by an isometry that keeps the tiling. Lattice tilings are tilings where the center points of all hypercubes of the same size form a lattice, therefore they are equitransitive.
In [Publ. Math. Debr. 59, No. 3--4, 317--326 (2001; Zbl 1006.52014)], \textit{A. Bölcskei} proved that Roger's filling is the only unilateral and equitransitive tiling of \(\mathbb R^3\) in cubes of two sizes. He also conjectured that in every dimension \(n\), there exists precisely one equitransitive unilateral tiling of \(\mathbb R^n\) by hypercubes of two sizes.
In the paper, a unilateral lattice tiling of \(\mathbb R^n\) into hypercubes of two different side lengths is constructed. This generalizes the Pythagorian tiling in \(\mathbb R^2\). It is shown that the constructed tiling is unique up to symmetries, proving a (weaker) variation of the Bölcskei conjecture.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Free and interfacial boundaries in individual-based models of multicellular biological systemshttps://zbmath.org/1526.920202024-02-15T19:53:11.284213Z"Germano, Domenic P. J."https://zbmath.org/authors/?q=ai:germano.domenic-p-j"Zanca, Adriana"https://zbmath.org/authors/?q=ai:zanca.adriana"Johnston, Stuart T."https://zbmath.org/authors/?q=ai:johnston.stuart-t"Flegg, Jennifer A."https://zbmath.org/authors/?q=ai:flegg.jennifer-a"Osborne, James M."https://zbmath.org/authors/?q=ai:osborne.james-mSummary: Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.Hadamard matrices related to a certain series of ternary self-dual codeshttps://zbmath.org/1526.940532024-02-15T19:53:11.284213Z"Araya, Makoto"https://zbmath.org/authors/?q=ai:araya.makoto"Harada, Masaaki"https://zbmath.org/authors/?q=ai:harada.masaaki"Momihara, Koji"https://zbmath.org/authors/?q=ai:momihara.koji\textit{G. Nebe} and \textit{D. Villar} provided a construction for ternary self-dual codes of length \(2(p+1)\) for all primes \(p\equiv 5\pmod{8}\), and they used this construction to find the third ternary extremal self-dual code of length 60; see [in: Optimal codes and related topics, OC 2013. Proceedings of the seventh international workshop, Albena, Bulgaria, September 6--12, 2013. Dedicated to Stefan Dodunekov (1945--2012). Sofia: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 158--163 (2013; Zbl 1433.94130)].
The present authors show that the ternary self-dual codes constructed in this way contain codewords forming Hadamard matrices of order \(2(p + 1)\) when \(p\equiv 5\pmod{24}\). The authors characterise these matrices and use this characterisation to show the codes are generated by the rows of the Hadamard matrices.
Finally, the authors show by way of a computer search that the third ternary extremal self-dual code of length 60 contains at least two inequivalent Hadamard matrices.
Reviewer: Thomas Britz (Sydney)Infinite families of 2-designs from linear codeshttps://zbmath.org/1526.940552024-02-15T19:53:11.284213Z"Du, Xiaoni"https://zbmath.org/authors/?q=ai:du.xiaoni"Wang, Rong"https://zbmath.org/authors/?q=ai:wang.rong"Tang, Chunming"https://zbmath.org/authors/?q=ai:tang.chunming"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qi.2Summary: Interplay between coding theory and combinatorial \(t\)-designs has attracted a lot of attention. It is well known that the supports of all codewords of a fixed Hamming weight in a linear code may hold a \(t\)-design. In this paper, we first settle the weight distributions of two classes of linear codes, and then determine the parameters of infinite families of 2-designs held in these codes.