Given two graphs G and H, an H-decomposition of G is a partition of the
edge set of G such that each part is either a single edge or forms a graph
isomorphic to H. Let f(n;H) be the smallest number  such that any graph
G of order n admits an H-decomposition with at most f(n;H)  parts. Pikhurko and
Sousa conjectured that f(n;H) = ex(n;H) for (H)  3 and all sufficiently
large n, where ex(n;H) denotes the maximum number of edges in a graph on n vertices not containing H as a subgraph. Their conjecture has been veried by
Ozkahya and Person for all edge-critical graphs H. In this article, the conjecture
is veried for the k-fan graph. The k-fan graph, denoted by F_k, is the graph on
2k + 1 vertices consisting of k triangles which intersect in exactly one common
vertex called the centre of the k-fan.

The process of industrialization – as a sustained trend in history and developing new socio-economic concepts – presuposes higher rates of growth of the economy and a structural change. It is on this basis diverse concepts of industrial revolutions have been accepted. These concepts have been coherent to reveal singular developments. Recently, the accepted concept of Industry 4.0 (i4.0) has shown some limitations regarding the need to develop automation technology in an anthropocentric orientation. This is why Industry 5.0 has two orientations: either the experiences that adopt this concept seek solutions to adapt the human factor to the features of the technology or, knowing the social and organizational requirements, look for solutions to develop this technology in accordance with these requirements. The debate is still developing. There is not yet conceptual maturity to propose a new topic that would be based in a new eventual structural change observed like a possible “Industry 6.0” era. Most arguments for an Industry 6.0 are still those which have been discussed for Industry 4.0, since they are mostly based on eventual not yet ready developments of technology. That is why we should continue the debate on the late developments of industrialization and its social and economic conditions.

The structural and chemical diversity of β-glucans is reflected on the variety of essential biological roles tackled by these polysaccharides. This natural heterogeneity requires an elaborate assortment of enzymatic mechanisms to assemble, degrade or modify, as well as to extract their full biotechnological potential. Recent metagenomic efforts have provided an unprecedented growth in potential new biocatalysts, most of which remain unconfirmed or uncharacterized. Here we report the first biochemical and structural characterization of two bacterial β-glucanases from the recently created glycoside hydrolase family 157 (LaGH157 and BcGH157) and investigate their molecular basis for substrate hydrolysis. Structural analysis by X-ray crystallography revealed that GH157 enzymes belong to clan GH-A, possessing a (β/α)8-barrel fold catalytic domain, two β-sandwich accessory domains and two conserved catalytic glutamates residues, with relative positions compatible with a retaining mechanism of hydrolysis. Specificity screening and enzyme kinetics suggest that the enzymes prefer mixed-linkage glucans over β-1,3-glucans. Activity screening showed that both enzymes exhibit pH optimum at 6.5 and temperature optimum for LaGH157 and BcGH157 at 25 °C and 48 °C, respectively. Product analysis with HPAEC-PAD and LC-MS revealed that both enzymes are endo-1,3(4)-β-glucanases, capable of cleaving β-1,3 and β-1,4-linked glucoses, when preceded by a β-1,3 linkage. Moreover, BcGH157 needs a minimum of 4 subsites occupied for hydrolysis to occur, while LaGH157 only requires 3 subsites. Additionally, LaGH157 possesses exohydrolytic activity on β-1,3 and branching β-1,6 linkages. This unusual bifunctional endo-1,3(4)/exo-1,3–1,6 activity constitutes an expansion on our understanding of β-glucan deconstruction, with the potential to inspire future applications.

The challenges of disciplinary integration in teaching practice, especially in the implementation of tasks that promote interdisciplinary approaches, highlight the importance of teachers' professional knowledge in this process. Using a naturalistic approach, this study aims to characterize the professional knowledge of a physics teacher when adopting an interdisciplinary approach using technology. The results of the research revealed that the teacher mobilizes different types of knowledge, showing a differentiated knowledge of the most appropriate mathematical application to teach a given piece of content, and knowledge of how pedagogical strategies can be aided through different applications using technology.

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This study aims to understand the functional thinking of 10th-grade students while studying functions. Specifically, we intend to answer the following research questions: what are the functional thinking processes used by 10th-grade students when studying functions? What difficulties do students present while learning functions? In view of the nature of this research objective, we adopted a qualitative and interpretative approach. In order to answer these questions, data were collected from the written records produced by the students while solving the proposed tasks, from records of the oral interactions during discussions and from a questionnaire. The results show that functional thinking processes were implicit in the resolution of the tasks proposed to the students. The students expressed an understanding of how the variables were related, presenting evidence of their functional thinking while working on the new concepts represented by the functions addressed in the proposed tasks. Some students expressed difficulties in interpreting the different types of representations associated with the functions, in retaining the necessary information from a graphical representation that would help them to draw conclusions and establish correspondences, in explaining functional relationships, and in interpreting the information provided by algebraic expressions. These difficulties can reduce the recognition of the relationships between variables and their behavior in the different representations, becoming an obstacle to learning for some students.

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The new book edited by Rodríguez and colleagues focuses on the topic of forecasting and responsible innovation. The original title is “Anticipación e Innovación Responsible: La construcción de futuros alternativos para la ciencia y la tecnologia” (Forecasting and Responsible Innovation: The construction of alternative futures for science and technology), and was published by Biblioteca Nueva, Madrid. Throughout this text, the reviewer is using the term forecasting instead of anticipation to convey the Spanish concept of “anticipación.” Both concepts are usually applied to “the act of looking forward” (Merriam-Webster dictionary1) or “the act of expecting or foreseeing something; expectation or presentiment” (Farlex free dictionary2) The concept of forecasting is usually used in scientific debate to mean “to estimate or predict in advance” (American Heritage Dictionary of the English Language, 2016) or “the process of making predictions based on past and present data and most commonly by analysis of trends” (Wikipedia3) (Glenn, 1994, p. 4) expressed this definition well by saying that “studying the future is not simply economic projections or sociological analysis or technological forecasting, but a multi-disciplinary examination of change in all major areas of life to find the interacting dynamics that are creating the next age.” The concept has been developed mainly by Armstrong (2001) and followed by Farrukh and Holgado (2020), Schnaars (2009), and Marinakis (2012), among others. The editors are professors and researchers from the University of Basque Country (EHU) and from the University of Mondragon (MU). The book involves a whole set of experts on the topic, including the editors themselves (Hannot Rodríguez, Sergio Urueña, Andoni Eizagirre, and Oier Imaz), and Armin Grunwald, René von Schomberg, Javier Garcia Fronti, Domingo García Marzá, Andoni Ibarra, and others. Although still published just in Spanish, it is an important contribution to the social sciences and philosophy of sciences regarding the analysis of alternative sociotechnical futures with strong ethical principles, which delineates an innovative approach in an era when the formation of public opinion largely suffers from systematic distortions based on vested interests.