Search Results (26)

Herein, a matrix representation of the Hamilton quaternion group by 4 × 4 square matrices with entries equal to −1, 0, or 1 is defined. It is proven that this group, denoted as $\left({\mathrm{Q}}_{\mathrm{M}},\circ \right)$, is a group of rotational symmetries of the four-dimensional hypercube ${\mathbb{Z}}_2^4,$ that is, a subgroup of the special orthogonal group ${\mathrm{SO}}_4\left(\mathbb{Z}\right).$ As a consequence, $\left({\mathrm{Q}}_{\mathrm{M}},\circ \right)$ is a group of rotational symmetries for each of the biological hypercubes RNY, YNY, YNR, and RNR. It is also proven that $\left({\mathrm{Q}}_{\mathrm{M}},\circ \right)$ is a group of permutations of the eight cubes contained in the four-dimensional hypercube ${\mathbb{Z}}_2^4.$ The latter is a novel result. It is also proven that the matrix quaternion group $\left({\mathrm{Q}}_{\mathrm{M}},\circ \right)$ is a normal subgroup of ${\mathrm{SO}}_4\left(\mathbb{Z}\right)$ and that the latter is a semidirect product of $\left({\mathrm{Q}}_{\mathrm{M}},\circ \right)$ with a copy of the special orthogonal group ${\mathrm{SO}}_3\left(\mathbb{Z}\right),$ also called an octahedral group because it is a group of rotational symmetries of a regular octahedron or of a three-dimensional cube. Full article

The aim of this paper is to introduce and study Zappa-Szép groupoids of inverse semigroups. Some properties of such kinds of groupoids are explored. As an application, an alternative proof of Billhardt’s $\lambda$-semidirect products is given. We finish with several examples that highlight the versatility and applicability of Zappa-Szép groupoids in various types of inverse semigroups. Full article

The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter $\lambda$ equals 0. We consider some related structures such as conformal modules, corresponding representations and $\mathcal{O}$-operators. Therefore, conformal derivations from Jacobi–Jordan conformal algebras to their conformal modules are used to describe conformal derivations of Jacobi–Jordan conformal algebras of the semidirect product type. Moreover, we study a class of Jacobi–Jordan conformal algebras called quadratic Jacobi–Jordan conformal algebras, which are characterized by mock-Gel’fand–Dorfman bialgebras. Finally, the $\mathbb{C}[\partial ]$-split extending structures problem for Jacobi–Jordan conformal algebras is studied. Furthermore, we introduce an unified product of a given Jacobi–Jordan conformal algebra J and a given $\mathbb{C}[\partial ]$-module K. This product includes some other interesting products of Jacobi–Jordan conformal algebras such as the twisted product and crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the $\mathbb{C}[\partial ]$-split extending structures problem. Full article

This review explores the current state of public key cryptography based on idempotent semirings, with an emphasis on tropical semirings. It examines key hard problems, such as the tropical discrete logarithm problem, semidirect tropical product problem, the factorization of tropical polynomials, and the matrix power function, that underpin the security of these protocols. Given the significant number of compromised protocols based on tropical semirings, most of which are variations of the Stickel protocol, we present three algorithms and classify schemes of this type. The analysis is further illustrated with a figure that maps the relationships between tropical Stickel’s-like protocols and the attacks targeting them. Additionally, the review provides an in-depth exploration of the vulnerabilities that have led to many tropical semiring-based cryptosystems being compromised. To address these challenges, the review highlights promising alternative approaches, including non-tropical idempotent platforms and non-idempotent options, such as supertropical semirings, which offer potential solutions to overcome known limitations. Furthermore, a discussion on the interplay between tropical cryptography and post-quantum cryptography is presented, raising the following question: what is the future of tropical cryptography? Full article

Mayotte is a small tropical island in the Comoros archipelago. It became recently a French department and much of its food, especially meat, is imported from abroad. The development of livestock farming is therefore a necessity. To understand the problems faced by Mahoran farmers, we organised semi-directive interviews with 15 farmers who reared cattle, sheep, goats or poultry. The first difficulty of farmers was limited access to land, especially for ruminants. This led to feed shortages. Another difficulty was the limited access to water and the poor quality of the roads to reach the farms. Poultry farmers were too dependent on importations of feed and laying hen or broiler genotypes from metropolitan France. The lack of organization for independent food productions (absence of abattoirs, cooperatives or organised markets) is also an obstacle to the development of the sector. Animal health, although not considered a major problem, has been a nuisance in the past (anthrax in cattle or salmonella in poultry). Mahoran farmers trust veterinarians or their assistants to manage health, although they complain about the high cost. Surprisingly, farmers use traditional medicine for many of their ailments, mostly based on local plants, but rarely for animals. Overall, our study reveals that larger land areas, better availability of money for investment and access to water and fodder are urgently required to improve livestock production and economic viability of farmers in Mayotte. Full article

Enhancing integrated crop–livestock systems (ICLSs) to improve land-use efficiency is a critical goal. Understanding the ICLS impacts on lowland soils is key to sustainable agricultural practices. Our objective was to test whether adopting ICLSs in lowlands improves soil structure, pore connectivity, and water and air permeability. This study was conducted in a long-term field trial, consisting of the following production systems with flood-irrigation rice: rice–fallow–rice, under conventional tillage and absence of grazing (RFR-ct); rice-grazed ryegrass–rice, under no-tillage and grazing (RGrR-nt); rice-grazed ryegrass–soybean-grazed ryegrass–rice, under no-tillage and grazing (RGrS/RGrR-nt); and a grazed pasture-consortium (winter) and succession field (summer), with no-till rice every 4 years (P4R-nt). Core samples were collected after grazing (October 2018), harvesting (March 2019), and grazing (October 2019). We analyzed soil air permeability, saturated hydraulic conductivity, pore connectivity by computed tomography. Soil tillage in a semi-direct system generated discontinuous porosity. Systems with intense trampling or less surface protection are affected by shearing on topsoil, reducing pore continuity. ICLSs are mainly composed of ryegrass–rice mitigated the harmful effects of trampling, and improved soil structure and functioning. Systems without soil tillage exhibited higher pore connectivity and pores with vertical orientation. Finally, soil tillage is not required to improve structural quality in ICLSs. Full article

Let $D\left(R_{m,n}\left(q\right)\right)$ be the Drinfeld double of Radford Hopf algebra $R_{m,n}\left(q\right)$ and $D\left(H_{s,t}\right)$ be the Drinfeld double of generalized Taft algebra $H_{s,t}$. Both $D\left(R_{m,n}\left(q\right)\right)$ and $D\left(H_{s,t}\right)$ have very symmetric structures. We calculate all Hopf automorphisms of $D\left(R_{m,n}\left(q\right)\right)$ and $D\left(H_{s,t}\right)$, respectively. Furthermore, we prove that the Hopf automorphism group $Au{t}_{Hopf}\left(D\left(R_{m,n}\left(q\right)\right)\right)$ is isomorphic to the direct sum ${\mathbb{Z}}_n⨁{\mathbb{Z}}_m$ of cyclic groups ${\mathbb{Z}}_m$ and ${\mathbb{Z}}_n$, the Hopf automorphism group $Au{t}_{Hopf}\left(D\left(H_{s,t}\right)\right)$ is isomorphic to the semi-direct products ${\mathbb{k}}^{*}⋊{\mathbb{Z}}_d$ of multiplicative group ${\mathbb{k}}^{*}$ and cyclic group ${\mathbb{Z}}_d$, where $s=td,{\mathbb{k}}^{*}=\mathbb{k}\backslash \left\{0\right\}$, and $\mathbb{k}$ is an algebraically closed field with char $\left(\mathbb{k}\right)\nmid t$. Full article

$U\left(n\right)$ is a semi-direct product group characterized by nontrivial homomorphisms mapping $U\left(1\right)$ into the automorphism group of $SU\left(n\right)$. For $U\left(3\right)$, there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of $U\left(3\right)$ Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a $\mathrm{mod}3$ permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same $U\left(3\right)$ quantum numbers. One can interpret this as three species of matter fields in the defining representation. Full article

In June 2020, a record-breaking Saharan dust storm, known as the “Godzilla” extreme event, caused significant dust transport from the Sahara Desert across the Atlantic Ocean to the United States. Based on satellite observations, the magnitude of aerosol optical depth (AOD) has consistently remained highest over the Atlantic Ocean for the past 18 years. This study uses satellite observations (including MODIS and CALIOP) and MERRA-2 reanalysis products to investigate the relationships between dust and marine clouds. During this extreme event, the concentration of AOD exhibits a synchronous anomaly with the cloud fraction (CF). Principal components analysis (PCA) results show that the enhanced temperature and specific humidity near the surface contribute the most to cloud development over the tropical Atlantic Ocean. Despite the reduced sensitivity of CF to aerosols, the semi-direct effect of dust can still play a crucial role during this extreme dust storm. We found that the presence of absorbing aerosols above the cloud layers warms the air, accompanied by an enhancement of surface moisture, thereby benefiting low-level cloud coverage. Full article

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and ${\alpha }^k$-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology. Full article

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ${\mathit{\ell }}_p\left({\mathbb{C}}^n\right)$ and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ${\mathit{\ell }}_p\left({\mathbb{C}}^n\right).$ Full article

This article focuses on the processes of territorialization of the local population’s living space, created by the governance regime in French Guiana, and their effects on the production of a tourist space in the context of sparsely populated regions. The Guiana Amazonian Park is analyzed as a territorialization agent with mechanisms that influence the development of tourism in the Maripasoula/Haut-Maroni zone. Our objective is to use the territorial framework to better understand the political and geographical dynamics that exist between the processes of the global production of tourist areas and those related to the local population’s management of the living space. Using Critical political geography framework, this study is based on documentary research and on 15 semi-directed interviews, conducted during a month-long stay in 2019, with different groups of stakeholders involved directly or indirectly in tourist activities. The paper first outlines the regional and local context of tourism in French Guiana. It also offers a territorial description of the different inclusion criteria for Sparsely Populated Regions in the Maripasoula/Haut-Maroni region, which is linked to the specific tourist practices in this territory. The processes of territorialization are then analyzed through the different governance regimes the French state created in order to understand how they fit into the production of a tourist space. Finally, a reflection on the future of tourism in this region is proposed, particularly regarding colonial governance regimes vis-à-vis Indigenous populations in the region. Our analysis demonstrates that tourism, along with any other form of activity to be developed in Maripasoula/Haut-Maroni territory, will be systematically confronted with the same structural constraints that have helped to reproduce the dynamics of territorial dispossession since the establishment of a colonial regime in the region. Full article

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the $deformed$ quaplectic group that is given by the semi-direct product of $U(1,3)$ with the $deformed$ (noncommutative) Weyl–Heisenberg group corresponding to $noncommutative$ fiber coordinates and momenta $[X_a,X_b]\ne 0$; $[P_a,P_b]\ne 0$. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks $[X_{a_1{a}_2\cdots a_n},X_{b_1{b}_2\cdots b_n}]\ne 0$; $[P_{a_1{a}_2\cdots a_n},P_{b_1{b}_2\cdots b_n}]\ne 0$. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting $x^{\mu },p^{\mu }$ operator variables (associated to an 8D curved phase space) to the canonical $Y^A,{\mathsf{\Pi }}^A$ operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the $classical$ limit, the embedding functions $Y^A(x,p),{\mathsf{\Pi }}^A(x,p)$ of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric $g_{\mu \nu }(x,p)$, the fiber metric of the vertical space $h^{ab}(x,p)$, and the nonlinear connection $N_{a\mu }(x,p)$ associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions. Full article

In this article, topologies on metagroups and quasigroups are studied. Topologies on smashed twisted wreath products of metagroups are scrutinized, which are making them topological metagroups. For this purpose, transversal sets are studied. As a tool for this, semi-direct products of topological metagroups are also investigated. They have specific features in comparison with topological groups because of the nonassociativity, in general, of metagroups. A related structure of topological metagroups is investigated. Particularly, their compact subloops and submetagroups are studied. Isomorphisms of topological unital quasigroups (i.e., loops) obtained by the smashed twisted wreath products are investigated. Examples are provided. Full article

Gastrointestinal nematodes (GINs) are a major health problem in tropical goat husbandry. The control of GIN has been nearly exclusively reliant on the use of anthelmintic treatments. Their wide use has provoked the appearance and diffusion of anthelmintic resistance. Therefore, there is a need to use anthelmintics only when they are really needed. This strategy of targeted selective treatment (TST) has been recommended. The selection of animals to be treated has been based either on the objective measures of GIN intensity (fecal nematode egg counts) performed in the laboratory or on indirect assessment such as anemia (FAMACHA©), diarrhea score or weight gains, particularly in sheep. The roughness of hair has also been proposed in goats. These indicators can be handled by the farmer. Their opinion on the importance of GINs, and the indicators that they are ready to accept and use have very rarely been studied. Goat for meat production is important in the French West Indies (especially in Guadeloupe) and GIN infection may significantly alter this production. Eighteen farmers participated in semi-directive interviews in order to appreciate their relation to goat GIN infection and the solutions they considered. Seventeen farms were investigated for fecal nematode egg counts, FAMACHA©, body score, and roughness of hair. The average infection by GINs was high (average fecal egg count 1562 and standard deviation 2028) with a wide range from one farm to another (from 0 to 25,000 eggs of GIN per gram of feces). The Haemonchus genera was predominant (54%), followed by Trichostrongylus (37%) and Oesophagostomum (9%). Young goats were less infected than adult goats since they were not yet grazing; males were more infected than females; and the Creole breed was more infected than the other breeds. Among the farming types, the professional ones were less infected compared with the traditional or mixed agriculture and husbandry farms. Those using targeted selective treatment did not have a significantly higher GIN infection than those treating the whole herd. Most of the characteristics were related and multivariate analysis could not match the intensity of GIN infection with any parameter. The frequency of anthelmintic treatments was negatively related to the use of body score, FAMACHA©, and hair roughness. The use of semi-directive interviews provided a wider understanding of the strategies and problems of farmers. The farmers valued their animals very much and diseases, in general, were a preoccupation, whereas parasites were not a major issue for traditional farmers. This is due to the important use of indicators and the belief in their value that gives comfort to the farmers that the parasites are being controlled. The extension services have well diffused the practice of indicators to the goat farmers of Guadeloupe, with some depending less on anthelmintics to control the gastrointestinal nematodes by using targeted selective treatments. Full article