In this paper, we consider the monoids of all endomorphisms, of all weak endomorphisms, of all strong endomorphisms and of all strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

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.

In this note, we aim to correct some of the results presented in [1]. Namely, the statements of Proposition 2.1, Corollary 2.2, Corollary 2.3, Theorem 2.4 and Theorem 2.6, concerning only the monoids OP_n and POP_n, have to exclude transformations of rank two. All other results of [1], as well as those mentioned above but for the monoids OR_n and POR_n, do not require correction.

[1] V.H. Fernandes, Oriented transformations on a finite chain: another description, Commun. Korean Math. Soc. 38 (2023), 725-731.

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

Political and economic crises and issues such as global warming, decreased fossil fuel reserves, environmental pollutions and acid rains are just some of the problems caused by improper energy consumption. Energy efficiency optimization with the right policy and definition of accurate scientific methods proven in energy consumption includes planning, enforcement, organizing permanent control, and continuously reviewing technology upgrades. Additionally, developing legal solutions to implement these programs prevents the destruction and waste of energy resources and reduces the side effects of improper energy consumption. The shape of the city and its components can affect the city’s quality and its spaces, as well as the air quality change, creating a particular climate in their area, causing phenomena such as heat islands, dust, rain, albedo, atmospheric inversion, and some others. By making changes in the shape of the city, both on a large scale and a micro-scale, atmospheric currents and, consequently, air quality in cities can be improved. Urban morphological indicators, including building height, building type, occupancy level, segmentation, block size, block shape, density, and urban green, can affect the weather and climate conditions outside and inside the building and, thus, the energy demand in the areas. By creating effective reforms in the model of configuration of the urban regions, including the pattern of land fragmentation, the physical characteristics of buildings, the network of passages and open spaces, it is possible to increase the energy efficiency of buildings in the long term in the regional(synoptic) climatic conditions of the city and the neighbourhood.

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities.

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.

In this paper we study the cyclic inverse monoid CI_n on a set Ω_n with n elements, i.e. the inverse submonoid of the symmetric inverse monoid on Ω_n consisting of all restrictions of the elements of a cyclic subgroup of order n acting cyclically on Ω_n. We show that CI_n has rank 2 (for n⩾2) and n⋅2^n−n+1 elements. Moreover, we give presentations of CI_n on n+1 generators and (n^2+3n+4)/2 relations and on 2 generators and (n^2−n+6)/2 relations. We also consider the remarkable inverse submonoid OCI_n of CI_n constituted by all its order-preserving transformations. We show that OCI_n has rank n and 3⋅2^n−2n−1 elements. Furthermore, we exhibit presentations of OCI_n on n+2 generators and (n^2+3n+8)/2 relations and on n generators and (n^2+3n)/2 relations.

In this note we give a presentation for the monoid IO_n of all order-preserving transformations of a n-chain whose ranges are intervals. We also consider the submonoid IO_n^- of IO_n consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.

In this paper, we consider the monoid DPW_n of all partial isometries of a wheel graph W_n with n+1 vertices. Our main objective is to determine the rank of DPW_n. In the process, we also compute the ranks of three notable subsemigroups of DPW_n. We also describe Green's relations of DPW_n and of its three considered subsemigroups.

In this paper we consider three submonoids of the dihedral inverse monoid DI_n, namely its submonoids OPDI_n, MDI_n and ODI_n of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank.

In this chapter, we examine the role of policies and other factors affecting digital technology (DT) integration in mathematics education. In particular, we develop a cross-national analysis of the impact on DT implementation in four countries: two countries in Europe (Italy and Portugal) and two countries in Latin America (Colombia and Mexico). We analyze the role that policies, political changes, reforms, curricula, educational organization and systems, sociocultural aspects, and teachers’ training, knowledge, and beliefs play toward possible DT implementations. We observe that there is a discourse in policies to promote digital technologies’ use, but in practice the availability and integration of such resources in mathematics classrooms is still scarce. We also note that the efforts done during the pandemic did not change this, promoting general ICT use, rather than DT resources that might enhance mathematics teaching and learning.

Iguanodontia is a diverse clade of herbivorous ornithischian dinosaurs that were speciose and abundant during the Jurassic and Cretaceous. Although the monophyly of Iguanodontia is well supported, their internal relationships have sparked heated debate due to several phylogenetic paradigm shifts. Late Jurassic basally branching iguanodontians in particular are not well understood in terms of their systematic affinities and evolutionary relevance. Their fossil record in Europe is meager compared with North America, with only a few species currently recognized. Two taxa are currently known from the Upper Jurassic of England, the basally branching styracosternan Cumnoria prestwichii and the putative dryosaurid Callovosaurus leedsi. In the Upper Jurassic of Portugal, the styracosternan Draconyx loureiroi and the dryosaurid Eousdryosaurus nanohallucis are presently the only described basally branching iguanodontians. Here we report a new species of early diverging iguanodontian from the Upper Jurassic Lourinhã Formation of western-central Portugal. The new species is clearly distinguished from all other coeval taxa by an exclusive combination of characters that include a tibia with a cnemial crest that is directed craniolaterally and a fibular condyle that is angled at 90° with respect to the proximal epiphysis, a fibula with symmetrical proximal margins, and a reduced metatarsal I. The phylogenetic relationships of the Lourinhã iguanodontian were explored using maximum parsimony and Bayesian inference. The two analyses recover the Lourinhã iguanodontian as an indeterminate dryomorphan, with more precise affinities precluded due to the current available material. Body size is estimated between 3 and 4 meters for the holotype specimen, adding to the diversity of small ornithopods already recognized in the paleoichnological record of the Lourinhã Formation. http://zoobank.org/urn:lsid:zoobank.org:pub:5F4D52D0-0F0B-4809-8561-BE58C7C97D45

The jigsaw-puzzle fit of South America and Africa is an icon of plate tectonics and continental drift. Fieldwork in Angola since 2002 allows the correlation of onshore outcrops and offshore geophysical and well-core data in the context of rift, sag, salt, and post-salt drift phases of the opening of the central South Atlantic. These outcrops, ranging in age from >130 Ma to <71 Ma, record Early Cretaceous outpouring of the Etendeka-Paraná Large Igneous Province (Bero Volcanic Complex) and rifting, followed by continental carbonate and siliciclastic deposition (Tumbalunda Formation) during the sagging of the nascent central South Atlantic basin. By the Aptian, evaporation of sea water resulted in thick salt deposits (Bambata Formation), terminated by sea floor spreading. The Equatorial Atlantic Gateway began opening by the early Late Cretaceous (100 Ma) and allowed flow of currents between the North and South Atlantic, creating environmental conditions that heralded the introduction of marine reptiles. These dramatic outcrops are a unique element of geoheritage because they arguably comprise the most complete terrestrially exposed geological record of the puzzle-like icon of continental drift.

Orange peels contain a considerable number of bioactive compounds such as carotenoids, that can be used as ingredients in high-value products. The aim of this study was to compare orange peel extracts obtained with different green solvents (vegetable oils, fatty acids, and deep eutectic solvents (DES)). In addition, the chemical characterization of a new hydrophobic DES formed by octanoic acid and l-proline (C8:Pro) was performed. The extracts were compared in terms of carotenoid extraction, antioxidant activity by three methods, color, and environmental impact. The results confirmed that the mixture of C8:Pro is a DES and showed the highest carotenoid extraction (46.01 µg/g) compared to hexane (39.28 µg/g). The antioxidant activity was also the highest in C8:Pro (2438.8 µM TE/mL). Finally, two assessment models were used to evaluate the greenness and sustainability of the proposed extractions. These results demonstrated the potential use of orange peels in the circular economy and industry.