Search Results (47)

This paper extends the notions of n-derivations and n-automorphisms from Lie algebras to nest algebras via exponential mappings. We establish necessary and sufficient conditions for triangularity, and examine the preservation of the radical, center, and ideals under these higher-order algebraic transformations. The induced group structures of n-automorphisms are explicitly characterized, including inner and non-abelian components. Several concrete examples demonstrate the applicability and depth of the theoretical findings. Full article

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures. 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

We describe the nonabelian exterior square $G\widehat{\wedge }G$ of a pro-p-group G (with p arbitrary prime) in terms of quotients of free pro-p-groups, providing a new method of construction of $G\widehat{\wedge }G$ and new structural results for $G\widehat{\wedge }G$. Then, we investigate a generalization of the probability that two randomly chosen elements of G commute: this notion is known as the “complete exterior degree” of a pro-p-group and we will use it to characterize procyclic groups. Among other things, we present a new formula, which simplifies the numerical aspects which are connected with the evaluation of the complete exterior degree. Full article

The neutrino sector offers one of the most sensitive probes of new physics beyond the Standard Model of Particle Physics (SM). The mechanism of neutrino mass generation is still unknown. The observed suppression of neutrino masses hints at a large scale, conceivably of the order of the scale of a rand unified theory (GUT), which is a unique feature of neutrinos that is not shared by the charged fermions. The origin of neutrino masses and mixing is part of the outstanding puzzle of fermion masses and mixings, which is not explained ab initio in the SM. Flavor model building for both quark and lepton sectors is important in order to gain a better understanding of the origin of the structure of mass hierarchy and flavor mixing, which constitute the dominant fraction of the SM parameters. Recent activities in neutrino flavor model building based on non-Abelian discrete flavor symmetries and modular flavor symmetries have been shown to be a promising direction to explore. The emerging models provide a framework that has a significantly reduced number of undetermined parameters in the flavor sector. In addition, such a framework affords a novel origin of $\mathcal{C}$$\mathcal{P}$ violation from group theory due to the intimate connection between physical $\mathcal{C}$$\mathcal{P}$ transformation and group theoretical properties of non-Abelian discrete groups. Model building based on non-Abelian discrete flavor symmetries and their modular variants enables the particle physics community to interpret the current and anticipated upcoming data from neutrino experiments. Non-Abelian discrete flavor symmetries and their modular variants can result from compactification of a higher-dimensional theory. Pursuit of flavor model building based on such frameworks thus also provides the connection to possible UV completions: in particular, to string theory. We emphasize the importance of constructing models in which the uncertainties of theoretical predictions are smaller than, or at most compatible with, the error bars of measurements in neutrino experiments. While there exist proof-of-principle versions of bottom-up models in which the theoretical uncertainties are under control, it is remarkable that the key ingredients of such constructions were discovered first in top-down model building. We outline how a successful unification of bottom-up and top-down ideas and techniques may guide us towards a new era of precision flavor model building in which future experimental results can give us crucial insights into the UV completion of the SM. 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

The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy $\mathit{H}\left(\gamma ,\mathcal{O}\right)$ is proportional to $\mathit{R}\left(\sigma \right)$. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop $\gamma$ embedded in a manifold $\mathcal{M}$, $\mathit{H}\left(\gamma ,\mathcal{O}\right)$ is an element of a Lie group $\mathcal{G}$; the curvature $\mathit{R}\left(\sigma \right)\in \mathfrak{g}$ is an element of the Lie algebra of $\mathcal{G}$. However, it turns out that the curvature form $\mathit{R}\left(\sigma \right)$ obtained from the small loop approximation is ambiguous, as the information of $\gamma$ and $\mathit{H}\left(\gamma ,\mathcal{O}\right)$ are insufficient for determining a specific plane $\sigma$ responsible for $\mathit{R}\left(\sigma \right)$. To resolve this ambiguity, it is necessary to specify the surface $\mathcal{S}$ enclosed by the loop $\gamma$; hence, $\sigma$ is defined as the limit of $\mathcal{S}$ when $\gamma$ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations. Full article

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. This depends on non-abelian simple groups with a subgroup of index pq being classified, where $p>q$ are odd primes. Full article

In this paper, we study the square of generalized Hamming graphs by the properties of abelian groups, and characterize some isomorphisms between the square of generalized Hamming graphs and the non-complete extended p-sum of complete graphs. As applications, we determine the eigenvalues of the square of some generalized Hamming graphs. Full article

In 1978, Friedberg and Lee introduced the phenomenological soliton bag model of hadrons, generalizing the MIT bag model developed in 1974 shortly after the formulation of QCD. In this model, quarks and gluons are confined due to coupling with a real scalar field $\rho$, which tends to zero outside some compact region $S\subset {\mathbb{R}}^3$ determined dynamically from the equations of motion. The gauge coupling in the soliton bag model runs as the inverse power of $\rho$, already at the semiclassical level. We show that this model arises naturally as a consequence of introducing the warped product metric $\mathrm{d}{s}_M^2+{\rho }^2\mathrm{d}{s}_G^2$ on the principal G-bundle $P(M,G)\cong M\times G$ with a non-Abelian group G over Minkowski space $M={\mathbb{R}}^{3,1}$. Confinement of quarks and gluons in a compact domain $S\subset {\mathbb{R}}^3$ is a consequence of the collapse of the bundle manifold $M\times G$ to M outside S due to shrinking of the group manifold G to a point. We describe the formation of such regions S as a dynamical process controlled by the order parameter field $\rho$. Full article

The factorization of groups into a Zappa–Szép product, or more generally into a k-fold Zappa–Szép product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group and has recently been applied to public-key cryptography. We provide a generalization of the k-fold Zappa–Szép product of cyclic groups, which we call $OGS$ decomposition. It is easy to see that the existence of an $OGS$ decomposition for all the composition factors of a non-abelian group G implies the existence of an $OGS$ for G itself. Since the composition factors of a soluble group are cyclic groups, it has an $OGS$ decomposition. Therefore, the question of the existence of an $OGS$ decomposition is interesting for non-soluble groups. The Jordan–Hölder theorem motivates us to consider the existence of an $OGS$ decomposition for finite simple groups. In 1993, Holt and Rowley showed that $PS{L}_2\left(q\right)$ and $PS{L}_3\left(q\right)$ can be expressed as a product of cyclic groups. In this paper, we consider an $OGS$ decomposition of $PS{L}_2\left(q\right)$ from a different point of view to that of Holt and Rowley. We look at its connection to the $BN$-$pair$ decomposition of the group. This connection leads to sequences over ${\mathbb{F}}_q$, which can be defined recursively, with very interesting properties, and are closely connected to Dickson and Chebyshev polynomials. Since every finite simple Lie-type group exhibits $BN$-$pair$ decomposition, the ideas in this paper might be generalized to further simple Lie-type groups. Full article

Total Coloring of a graph G is a type of graph coloring in which any two adjacent vertices, an edge, and its incident vertices or any two adjacent edges do not receive the same color. The minimum number of colors required for the total coloring of a graph is called the total chromatic number of the graph, denoted by ${\chi }^{″}\left(G\right)$. Mehdi Behzad and Vadim Vizing simultaneously worked on the total colorings and proposed the Total Coloring Conjecture (TCC). The conjecture states that the maximum number of colors required in a total coloring is $\Delta \left(G\right)+2$, where $\Delta \left(G\right)$ is the maximum degree of the graph G. Graphs derived from the symmetric groups are robust graph structures used in interconnection networks and distributed computing. The TCC is still open for the circulant graphs. In this paper, we derive the upper bounds for ${\chi }^{″}\left(G\right)$ of some classes of Cayley graphs on non-abelian groups, typically Cayley graphs on the symmetric groups and dihedral groups. We also obtain the upper bounds of the total chromatic number of complements of Kneser graphs. Full article

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is $\mathbb{Z}\times {\mathbb{S}}^1$, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group $\mathbb{Z}\times \mathrm{SO}\left(2\right)$. In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on $L^2\left({\mathbb{S}}^1\right)$, is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress. Full article

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph ${\mathrm{\Gamma }}_{\mathcal{G}}$ of a finite group $\mathcal{G}$ is a graph where non-central elements of $\mathcal{G}$ are its vertex set, while two different elements are edge connected when they do not commute in $\mathcal{G}$. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of $SL(2,\mathbb{C})$. We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups. Full article

Let $a\left(n\right)$ be the number of non-isomorphic abelian groups of order n. In this paper, we study a symmetric form of the average value with respect to $a\left(n\right)$ and prove an asymptotic formula. Furthermore, we study an analogue of the well-known Titchmarsh divisor problem involving $a\left(n\right).$ Full article