Search Results (90)

The overlap function, a continuous aggregation function, is widely used in classification, decision-making, image processing, etc. Compared to applications, overlap functions have also achieved fruitful results in theory, such as studies on the fundamental properties of overlap functions, various generalizations of the concept, and the construction of additive and multiplicative generators based on overlap functions. However, most of the research studies on the overlap functions mentioned above assume commutativity and continuity, which can limit their practical applications. In this paper, we remove the symmetry and continuity from overlap functions and define discrete pseudo-quasi-overlap functions on finite chains. Meanwhile, we also discuss their related properties. Then, we introduce pseudo-quasi-overlap functions on sub-chains and construct discrete pseudo-quasi-overlap functions on finite chains using these sub-chain functions. Unlike quasi-overlap functions on finite chains generated by the ordinal sum, discrete pseudo-quasi-overlap functions on finite chains constructed through pseudo-quasi-overlap functions on different sub-chains are dissimilar. Eventually, we remove the continuity from pseudo-automorphisms and propose the concept of pseudo-quasi-automorphisms. Based on this, we utilize pseudo-overlap functions, pseudo-quasi-automorphisms, and integral functions to obtain discrete pseudo-quasi-overlap functions on finite chains; moreover, we apply them to fuzzy multi-attribute group decision-making. The results indicate that—compared to overlap functions and pseudo-overlap functions—discrete pseudo-quasi-overlap functions on finite chains have stronger flexibility and a wider range of practical applications. Full article

Let $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$ be the moduli space of $Spin(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface X of genus $g\ge 2$. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of $Spin(8,\mathbb{C})$ acts on $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$, and those Higgs bundles that admit a reduction in the structure group to $G_2$ are fixed points of this action. This defines a map of moduli spaces of Higgs bundles $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$. In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with $\mathcal{M}(Spin(8,\mathbb{C}\left)\right)$. In particular, it is checked that the map $\mathcal{M}\left(G_2\right)\to \mathcal{M}(Spin(8,\mathbb{C}))$ is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of $G_2$ and $Spin(8,\mathbb{C})$-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved. Full article

In this paper, we introduce two-term differential $Lei{b}_{\infty }$-conformal algebras and give characterizations of some particular classes of such two-term differential $Lei{b}_{\infty }$-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras. Full article

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. Full article

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving. Full article

This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of idempotent transformations with restricted range, revealing that these automorphisms are inner ones and induced by the units of S. Additionally, we establish that the automorphism group $\mathrm{Aut}\left(S\right)$ is isomorphic to $U_S$, the group of units of S. These findings extend previous results on semigroups of transformations, enhancing their applicability and providing a more unified theory. The practical implications of this work span multiple fields, including automata theory, coding theory, cryptography, and graph theory, offering tools for more efficient algorithms and models. By simplifying complex concepts and providing a solid foundation for future research, this study makes significant contributions to both theoretical and applied mathematics. 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 this paper, we deduce some hyperstability results for a generalized class of homogeneous Pexiderized functional equations, expressed as ${\sum }_{\rho \in \Gamma }f\left(x\rho .y\right)=\mathsf{\ell }f\left(x\right)+\mathsf{\ell }g\left(y\right)$, $x,y\in M$, which is inspired by the concept of Ulam stability. Indeed, we prove that function f that approximately satisfies an equation can, under certain conditions, be considered an exact solution. Domain M is a monoid (semigroup with a neutral element), $\Gamma$ is a finite subgroup of the automorphisms group of M, ℓ is the cardinality of $\Gamma$, and $f,g:M\to G$ such that $(G,+)$ denotes an ℓ-cancellative commutative group. We also examine the hyperstability of the given equation in its inhomogeneous version ${\sum }_{\rho \in \Gamma }f\left(x\rho .y\right)=\mathsf{\ell }f\left(x\right)+\mathsf{\ell }g\left(y\right)+\psi (x,y),x,y\in M,$ where $\psi :M\times M\to G$. Additionally, we apply the main results to elucidate the hyperstability of various functional equations with involutions. Full article

A graph is symmetric if its automorphism group is transitive on the arcs of the graph. Guo et al. determined all of the connected seven-valent symmetric graphs of order $8p$ for each prime p. We shall generalize this result by determining all of the connected seven-valent symmetric graphs of order $8pq$ with p and q to be distinct primes. As a result, we show that for each such graph of $\mathsf{\Gamma }$, it is isomorphic to one of seven graphs. Full article

Here, we study Frobenius bimodules associated with a pair of automorphisms of an algebra and discuss their basic properties. In particular, some equivalent conditions for a finite-dimensional bimodule are proved to be Frobenius and some isomorphisms between Ext-groups and Tor-groups of Frobenius modules over finite dimensional algebras are established. Full article

The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{a_i\right\}$ with n punctures $a_i$ are encoded in its fundamental group ${\pi }_1\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_1$ over a Lie group G as globalized by the character variety $\mathcal{C}=\mathrm{Hom}({\pi }_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere $\mathsf{\Sigma }=S_2^{\left(4\right)}$ and the ‘space-time-spin’ group $G=S{L}_2\left(\mathbb{C}\right)$. In such a situation, $\mathcal{C}$ possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or $P_{VI}$); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of $P_{VI}$. In this paper, we feature the parametric representation of some solutions of $P_{VI}$: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups $f_p$ encountered in TQC or DNA/RNA sequences are proposed. Full article

Let ${\left(k_{\mu }\right)}_{\mu =1}^4$ be a quartet of cyclic cubic number fields sharing a common conductor $c=pqr$ divisible by exactly three prime(power)s, $p,q,r$. For those components of the quartet whose 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic, the automorphism group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ of the maximal metabelian unramified 3-extension of $k_{\mu }$ is determined by conditions for cubic residue symbols between $p,q,r$ and for ambiguous principal ideals in subfields of the common absolute 3-genus field $k^{*}$ of all $k_{\mu }$. With the aid of the relation rank $d_2\left(\mathfrak{M}\right)$, it is decided whether $\mathfrak{M}$ coincides with the Galois group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })$ of the maximal unramified pro-3-extension of $k_{\mu }$. Full article

This paper contributes to the classification of flag-transitive symmetric 2-$(v,k,\lambda )$ designs with $\lambda$ prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is $PS{L}_n\left(q\right)$. Full article

Consider G to be a finite group and p to be a prime divisor of the order $\left|G\right|$ in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro’s vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let $\varphi \in Irr\left(G\right)$ have a positive height, and let there be an anchor group $A_{\varphi }$. We prove that if the normalizer $N_G\left(A_{\varphi }\right)$ is p-constrained, then $O_{\acute{p}}\left(N_G\left(A_{\varphi }\right)\right)\ne \left\{1_G\right\}$, where $O_{\acute{p}}\left(N_G\left(A_{\varphi }\right)\right)$ is the maximal normal $\acute{p}$ subgroup of $N_G\left(A_{\varphi }\right)$. We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results. Full article