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Erratum and addendum to "On the Isomorphism Classes of Transversals III", J. Indian Math. Soc., Vol. 82(3-4), (2015), 83-96. [ABSTRACT FROM AUTHOR]

Given a collection of hypergraphs H=(H1,...,Hm)${{\bf H}}=(H_1, \ldots , H_m)$ with the same vertex set, an m$m$‐edge graph F⊂∪i∈[m]Hi$F\subset \cup _{i\in [m]}H_i$ is a transversal if there is a bijection ϕ:E(F)→[m]$\phi :E(F)\rightarrow [m]$ such that e∈E(Hϕ(e))$e\in E(H_{\phi (e)})$ for each e∈E(F)$e\in E(F)$. How large does the minimum degree of each Hi$H_i$ need to be so that H${{\bf H}}$ necessarily contains a copy of F$F$ that is a transversal? Each Hi$H_i$ in the collection could be the same hypergraph, hence the minimum degree of each Hi$H_i$ needs to be large enough to ensure that F⊆Hi$F\subseteq H_i$. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac‐type results for (powers of) Hamilton cycles. For example, we derive that any collection of rn$rn$ graphs on an n$n$‐vertex set, each with minimum degree at least (r/(r+1)+o(1))n$(r/(r+1)+o(1))n$, contains a transversal copy of the r$r$th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture. [ABSTRACT FROM AUTHOR]

The research results published so far have shown that success in mathematics positively affects the retention rate at technical and economic universities. Various studies have explored different forms, methods, and tools in mathematics teaching to improve student success, but an effective teaching model is yet to be identified. Based on previous experience and published results, a model of effective teaching was proposed with an emphasis on developing selected cross-sectional competencies to increase student retention. A recent study aimed to verify the effectiveness of this model in the pilot subject Mathematics I. The study included 211 first-year students of 7 bachelor's technical-economic study programs. Results showed significant improvement in the group of students who struggled with mathematics. The study confirmed that an appropriately chosen intervention in mathematics can increase the success rate of completing technical and economic study programs, and this approach can also be applied at other universities. [ABSTRACT FROM AUTHOR]

In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg's representation of the group G=\text {PSO}(2n+2) and a Hitchin fibration. Consistent with the result over \mathbb {Q} of Arul Shankar and Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely average number 6. [ABSTRACT FROM AUTHOR]

The transversal number Τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard [Discuss. Math. Graph Theory, 22 (2002), pp. 199-210] conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then Τ(H) < 1/4n. In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture. [ABSTRACT FROM AUTHOR]

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This cross-sectional study investigates mathematical and mathematics educational values of students across grade levels in Turkey. The sample of the study consisted of 970 students, including 511 fifth-grade and 459 ninth-grade students who received education in seven geographical regions of Turkey which were determined via a proportional stratified random sampling method. The data of the study were obtained with a bipolar scale developed within the scope of the What I Find Important [WIFI] (in maths learning) project. The results revealed a statistically significant difference in the value pairs of process–product, pleasure–hardship, application–computation, facts–ideas, exposition–exploration, recalling–creating, rationalism–objectism, whereas results revealed no statistically significant difference in terms of value pairs of ability–effort, openness–mystery and control–progress. Additionally, it was determined that at the end of middle school, the pleasure–hardship value pair evolved into hardship value, the application–computation value pair evolved into computation value and finally the exposition–exploration value pair evolved into exposition value. [ABSTRACT FROM AUTHOR]

We consider the point-to-point log-gamma polymer of length 2N in a half-space with i.i.d. Gamma-1(2θ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\text {Gamma}}^{-1}(2\theta )$$\end{document} distributed bulk weights and i.i.d. Gamma-1(α+θ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\text {Gamma}}^{-1}(\alpha +\theta )$$\end{document} distributed boundary weights for θ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta >0$$\end{document} and α>-θ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >-\theta $$\end{document}. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when α=N-1/3μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha =N^{-1/3}\mu $$\end{document} for μ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu \in \mathbb {R}$$\end{document} fixed (critical regime) and when α>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >0$$\end{document} is fixed (supercritical regime). In particular, in these two regimes, we show that after appropriate centering, the free energy process with spatial coordinate scaled by N2/3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{2/3}$$\end{document} and fluctuations scaled by N1/3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{1/3}$$\end{document} is tight. These regimes correspond to a polymer measure which is not pinned at the boundary. This is the first instance of establishing the 2/3 transversal exponent for a positive temperature half-space model, and the first instance of the 1/3 fluctuation exponent besides precisely at the boundary where recent work of Imamura et al. (Solvable models in the KPZ class: approach through periodic and free boundary Schur measures. arXiv:2204.08420. 2022) applies and also gives the exact one-point fluctuation distribution (our methods do not access exact fluctuation distributions). Our proof relies on two inputs—the relationship between the half-space log-gamma polymer and half-space Whittaker process (facilitated by the geometric RSK correspondence as initiated in Corwin et al. (Duke Math J 163(3):513–563, 2014), O’Connell et al. (Invent Math 197(2):361–416, 2014), and an identity in Barraquand and Wang (Int Math Res Not 2023:11877, 2022) which relates the point-to-line half-space partition function to the full-space partition function for the log-gamma polymer. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated in Corwin and Hammond (Invent Math 195(2):441–508, 2014), a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. [ABSTRACT FROM AUTHOR]

Competence-based education is a widely implemented educational approach, but more research is needed into the relationship between transversal competences and individual disciplines. In this article, we present the results of a study focusing on how the transversal competence of multiliteracy is contextually defined in Finnish local curricula in the disciplines of mathematics and social studies. The article offers new insights into the discussion between content- and competence-based educations by introducing the concept of disciplinary contextualisation. Based on the qualitatively analysed data, four different types of disciplinary contextualisation are presented and further discussed. The study also engages with the discussion in the field of multiliteracy by offering insights into the ways in which multiliteracy is rationalised, defined, and developed in the analysed disciplines. Multiliteracy contextualisations share features, but also differences, between the disciplines, illustrating the importance of taking into account the disciplinary perspective when discussing the development of competences in basic education. [ABSTRACT FROM AUTHOR]

The aim of this note is to give an elementary proof of the following fact: given three red convex sets and three blue convex sets in I E 3 , such that every red intersects every blue, there is a line transversal to the reds or there is a line transversal to the blues. This is a special case of a theorem of Montajano and Karasev (Discrete Comput Geom 46(2):283–300, 2011) and generalizes, in a sense, the colourful Helly theorem due to Lovász (cf. Discrete Math 40(2,3):141–152, 1982) [ABSTRACT FROM AUTHOR]

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K. Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233–245) to general transversely symplectic foliations. [ABSTRACT FROM AUTHOR]

We discuss centroaffine geometry of polygons in 3-space. For a polygon X that is locally convex with respect to an origin together with a transversal vector field U, we define the centroaffine dual pair (Y, V) similarly to Nomizu and Sasaki (Nagoya Math J 132:63–90, 1993). We prove that vertices of (X, U) correspond to flattening points for (Y, V) and also that constant curvature polygons are dual to planar polygons. As an application, we give a new proof of a known four flattening points theorem for spatial polygons. [ABSTRACT FROM AUTHOR]

Extended formulations of (k , l) -sparsity matroids defined on graphs with n vertices and m edges are investigated by Iwata et al. [Extended formulations for sparsity matroids, Math. Program. 158 (2016), pp. 565–574]. This note interprets results on (k , l) -lower-truncated transversal polymatroids by the first author in 1983, from the viewpoint of extended formulations, and shows the same O (m n) bound when k ≥ l and a better bound O (m 2) when k

A subset T of vertices in a hypergraph H is a transversal if T has a nonempty intersection with every edge of H. The transversal number τ (H) of H is the minimum size of a transversal in H. A hypergraph H is 3-uniform if every edge of H has size 3. Let H be a 3-uniform hypergraph with n H vertices and m H edges. Tuza (Discrete Math 86:117–126, 1990) and Chvátal and McDiarmid (Combinatorica 12:19–26, 1992) showed that 4 τ (H) ≤ n H + m H . Chvátal and McDiarmid also showed that 6 τ (H) ≤ 2 n H + m H . The linear hypergraphs achieving equality in these bounds were characterized by the authors (Henning and Yeo in J Graph Theory 59:326–348, 2008; Discrete Math 313:959–966, 2013). In this paper, we show that these bounds can be improved if we impose some structural properties on H. We show that if H does not contain a subhypergraph isomorphic to the affine plane AG(2, 3) of order 3 with two vertices deleted, then 17 τ (H) ≤ 5 n H + 3 m H . The total domination number γ t (G) of a graph G is the minimum cardinality of a set S of vertices so that every vertex in G is adjacent to some vertex in S. It is known (Archdeacon et al. in J Graph Theory 46:207–210, 2004) that if G is a graph of order n with minimum degree at least 3, then γ t (G) ≤ 1 2 n , and that this bound is tight. As a consequence of our hypergraph results, we show that if G is a graph of order n with minimum degree at least 3 that contains no 4-cycles and no specified graph on 12 vertices as a subgraph, then γ t (G) ≤ 8 17 n . [ABSTRACT FROM AUTHOR]

In this article, the author as the President of the Royal Spanish Mathematical Society (RSME) send a message of solidarity to the families of the colleagues and colleagues from the RSME who have been affected by COVID-19. It mentions about establishing transversal lines of action given the confluence of themes, and define dynamization actions and accountability of the commissions.

Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of $$\mathbb {D}^2\times \mathbb {R}^n$$ equals $$\pi /\sqrt{12}$$ for all natural numbers n. [ABSTRACT FROM AUTHOR]

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, 'triple products' of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish. [ABSTRACT FROM AUTHOR]