Search Results (9)

We study fiber bundles where the fibers are not a group G but a free G-space with disjoint orbits. The fibers are then not torsors but disjoint unions of these; hence, we like to call them semi-torsors. Bundles of semi-torsors naturally generalize principal bundles, and we call these semi-principal bundles. These bundles admit parallel transport in the same way that principal bundles do. The main difference is that lifts may end up in another group orbit, meaning that the change cannot be described by group translations alone. The study of such effects is facilitated by defining the notion of a basis of a G-set, in analogy with a basis of a vector space. The basis elements serve as reference points for the orbits so that parallel transport amounts to reordering the basis elements and scaling them with the appropriate group elements. These two symmetries of the bases are described by a wreath product group. The notion of basis also leads to a frame bundle, which is principal and so allows for a conventional treatment. In fact, the frame bundle functor is found to be a retraction from the semi-principal bundles to the principal bundles. The theory presented provides a mathematical framework for a unified description of geometric phases and exceptional points in adiabatic quantum mechanics. Full article

Quality assurance through visual inspection plays a pivotal role in agriculture. In recent years, deep learning techniques (DL) have demonstrated promising results in object recognition. Despite this progress, few studies have focused on assessing human visual inspection and DL for defect identification. This study aims to evaluate visual human inspection and the suitability of using DL for defect identification in products of the floriculture industry. We used a sample of defective and correct decorative wreaths to conduct an attribute agreement analysis between inspectors and quality standards. Additionally, we computed the precision, accuracy, and Kappa statistics. For the DL approach, a dataset of wreath images was curated for training and testing the performance of YOLOv4-tiny, YOLOv5, YOLOv8, and ResNet50 models for defect identification. When assessing five classes, inspectors showed an overall precision of 92.4% and an accuracy of 97%, just below the precision of 93.8% obtained using YOLOv8 and YOLOv5 with accuracies of 99.9% and 99.8%, respectively. With a Kappa value of 0.941, our findings reveal an adequate agreement between inspectors and the standard. The results evidence that the models presented a similar performance to humans in terms of precision and accuracy, highlighting the suitability of DL in assisting humans with defect identification in artisanal-made products from floriculture. Therefore, by assisting humans with digital technologies, organizations can embrace the full potential of Industry 4.0, making the inspection process more intelligent and reliable. Full article

This study provides a scholarly examination of fundamental concepts within the field of group theory, specifically focusing on topics such as the wreath product and powerful p-groups. We examine the characteristics pertaining to the structure of the wreath product of cyclic p-groups, with a specific focus on the groups that are powerfully embedded within it. The primary discovery pertains to the construction of the powerful wreath product and the quasi-powerful wreath product. In this study, we establish that subgroups are powerful within the wreath product, specifically focusing on p-groups. The aforementioned outcome is derived from the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H. 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

We obtain a large number of degree and distance-based topological indices, graph and Laplacian spectra and the corresponding polynomials, entropies and matching polynomials of n-dimensional hypercubes through the use of Hadamard symmetry and recursive dynamic computational techniques. Moreover, computations are used to provide independent numerical values for the topological indices of the 11- and 12-cubes. We invoke symmetry-based recursive Hadamard transforms to obtain the graph and Laplacian spectra of nD-hypercubes and the computed numerical results are constructed for up to 23-dimensional hypercubes. The symmetries of these hypercubes constitute the hyperoctahedral wreath product groups which also pave the way for the symmetry-based elegant computations. These results are used to independently validate the exact analytical expressions that we have obtained for the topological indices as well as graph, Laplacian spectra and their polynomials. We invoke a robust dynamic programming technique to handle the computationally intensive generation of matching polynomials of hypercubes and compute all matching polynomials up to the 6-cube. The distance degree sequence vectors have been obtained numerically for up to 108-dimensional cubes and their frequencies are found to be in binomial distributions akin to the spectra of n-cubes. Full article

Let $\mathbb{C}{\mathsf{A}}_n=\mathbb{C}[S_2\wr S_2\wr \cdots \wr S_2]$ be the group algebra of an n-step iterated wreath product. We prove some structural properties of ${\mathsf{A}}_n$ such as their centers, centralizers, and right and double cosets. We apply these results to explicitly write down the Mackey theorem for groups ${\mathsf{A}}_n$ and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of ${\displaystyle \underset{m\ge 0}{⨁}({\mathsf{A}}_m,{\mathsf{A}}_n)-}$bimodules. A complete description of the category is an open problem. Full article

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused the global pandemic, coronavirus disease-2019 (COVID-19) which has resulted in 60.4 million infections and 1.42 million deaths worldwide. Mathematical models as an integral part of artificial intelligence are designed for contact tracing, genetic network analysis for uncovering the biological evolution of the virus, understanding the underlying mechanisms of the observed disease dynamics, evaluating mitigation strategies, and predicting the COVID-19 pandemic dynamics. This paper describes mathematical techniques to exploit and understand the progression of the pandemic through a topological characterization of underlying graphs. We have obtained several topological indices for various graphs of biological interest such as pandemic trees, Cayley trees, Christmas trees, and the corona product of Christmas trees and paths. We have also obtained an analytical expression for the thermodynamic entropies of pandemic trees as a function of R⁰, the reproduction number, and the level of spread, using the nested wreath product groups. Our plots of entropy and logarithms of topological indices of pandemic trees accentuate the underlying severity of COVID-19 over the 1918 Spanish flu pandemic. Full article

The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group $A_{2^k}$ is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of $A_{2^k}$ is constructed. It is shown that ${\left(Sy{l}_2{A}_{2^k}\right)}^2=Sy{l}_2^{\prime }{A}_{2^k},k>2$ . It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i},p_i\in \mathbb{N}$ equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width $\left(cw\right(G\left)\right)$ of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups $Sy{l}_2\left(A_{2^k}\right)$ of $A_{2^k}$ is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group $A_{2^k}$ , where $k>2$ , the permutation group $S_{2^k}$ , as well as Sylow p-subgroups of $Sy{l}_2{A}_{p^k}$ as well as $Sy{l}_2{S}_{p^k}$ ) are equal to 1 was obtained. A commutator width of permutational wreath product $B\wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B\wr C_n$ for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP. Full article