Asian-European Journal of Mathematics

In 1904, Schur proved his famous result which says that if the central factor group of a given group is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, $G/L(G)$, is finite, then so is its autocommutator subgroup, $K(G)$. In the present paper, for a given automorphism $\alpha$ of the group $G$, we introduce the concept of left $2_{\alpha }$-Engel, $L_2^{\alpha }(G)$, and $2_{\alpha }$-Engel commutator, $K_2^{\alpha }(G)$. Then under some condition, we prove that the finiteness of $G/L_2^{\alpha }(G)$ implies that $K_2^{\alpha }(G)$ is also finite. We also construct an upper bound for the order of $K_2^{\alpha }(G)$ in terms of the order of $G/L_2^{\alpha }(G)$.

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

In this paper, we prove some fixed point theorems for a new type of generalized contractive mappings involving $C$-class function, ${\alpha }_s^{\delta }$-admissible type mapping and Suzuki-type mappings in the frame work of complete $G_b$-metric spaces. The results obtained in this work generalize and improve some fixed point results in the literature.

Given a positive integer $q$, in this paper, we investigate some more properties of the $q$-capability of groups. For instance, the relationship between $q$-capability and the varietal capability is determined. Moreover, we introduce the notion of $q$-epicenter for a group and then we obtain some criteria for the $q$-capability of groups. Finally, as an application, we characterize all $q$-capable extra-special $p$-groups when $q$ is a power of $p.$

Here, we characterize the linear operators that preserve factor rank of matrices over additively-idempotent multiplicatively-cancellative semirings. The main results in this paper generalize the corresponding results on the two element Boolean algebra [L. B. Beasley and N. J. Pullman, Boolean-rank-preserving opeartors and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984) 55–77] and on the max algebra [R. B. Bapat, S. Pati and S.-Z. Song, Rank preservers of matrices over max algebra, Linear Multilinear Algebra 48(2) (2000) 149–164]; and hold on max-plus algebra and some other tropical semirings.

This paper is devoted to initial value problems for impulsive fractional differential equations of Caputo–Fabrizio type fractional derivative. By means of Banach’s fixed point theorem and Schaefer’s fixed point theorem, the existence and uniqueness results are obtained. Finally, an example is given to illustrate one of the main results.

In this paper, we have proved that minimum condition in a pseudo-ordered set (psoset) is equivalent to the descending pseudo-chain condition. Characterization of a pseudo-chain in an acyclic psoset is obtained using graph theoretic approach as transitivity need not hold in psosets. Skala proved that every complete trellis has Fixed Point Property. A counterexample given in this paper shows that trellis having Fixed Point Property need not be complete. The notion of weakly isotone mapping and Strong Fixed Point Property is introduced in a psoset and the characterization of weakly isotone mapping is obtained. It is proved that a connected psoset containing a nontrivial cycle does not have Strong Fixed Point Property.

Let $F_q$ be a finite field with characteristic $p>0$ having $q=p^n$ elements and $D_{60}$ be the dihedral group of order $60$. In this paper, we have obtained the structure of unit groups of group algebra $F_q{D}_{60}$, for any prime $p>0$.

In this paper, the Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are obtained by using the recurrence relation satisfied by them. Then, some linear and quadratic relations are given between Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials.

In this work, we prove Clarkson-type and Nash-type inequalities in the Dunkl setting on ${ℝ}^d$ for $L^p$-functions. By combining these inequalities, we show Heisenberg-type inequalities for the Dunkl transform on ${ℝ}^d$, and we deduce local-type uncertainty inequalities for the Dunkl transform on ${ℝ}^d$.

We define $f:E(G)\to \{1,2,\dots ,k_e\}$ and $f:V(G)\to \{0,2,\dots ,2k_v\}$ for $k_e,k_v\in N\cup \{0\}$, where $k=max\{k_e,2k_v\}$. The labeling $f$ is called a vertex irregular reflexive $k$-labeling if any vertices have distinct weight which $w_f(u)\ne w_f(v)$ for any vertices in a graph $G$. The weight of a vertex $u\in V(G)$ is defined as the sum of the labels of vertex and the labels of all edges incident this vertex. The smallest $k$ for which such labeling exists is called reflexive vertex strength of $G$, denoted by $\mathrm{\text{rvs}}(G)$. In this paper, we determine the reflexive vertex strength of gear graphs, book graphs, triangular book graph and the disjoint union of gear graphs.

The main purpose of this paper is to describe the structure of a pair of additive mappings that are commuting on a semiprime ring. Furthermore, we prove that the existence of different commuting epimorphisms on a prime ring forces the ring to be commutative. Finally, we characterize additive mappings, which act as homomorphisms or antihomomorphisms on a semiprime ring.

We investigate the parametrizations of loxodromes on the time-like rotational surfaces by using suitable Lorentzian angles in Minkowski space-time. Also, some examples and corresponding graphs are given by using Mathematica.

In this paper, we prove the boundedness for the maximal and fractional maximal operators and Riesz potential-type operator associated with the Kontorovich–Lebedev transform (KL transform)in the $L^p({ℝ}_{+},x^{-\beta }dx)$ spaces.

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The general reduced second Zagreb index of $G$ is defined as ${\mathrm{\text{GRM}}}_{\beta }(G)={\sum }_{uv\in E(G)}(d(u)+\beta )(d(v)+\beta )$, where $\beta$ is any real number and $d(v)$ is the degree of the vertex $v$ of $G$. In this paper, the general reduced second Zagreb index of the Cartesian product, corona product, join of graphs and two new operations of graphs are computed.

Approximation formulas for the Genocchi polynomials of complex order are obtained using contour integration with the contour avoiding branch cuts. An alternative expansion is also obtained by expanding a function involving the generating function in a two-point Taylor expansion.

Using the Dziok–Srivastava operator, we describe the class of p-valent analytical functions ${\varkappa }_p^{q,s}(k,{\alpha }_1,\alpha ,\beta ,\mu ,\lambda )$ and we obtain certain interesting properties of this class.

Based on a state feedback controller, a new 6D hyperchaotic system with real variables and a self-excited attractor is constructed. The dynamic behavior of the new system is investigated in terms of Lyapunov exponents, equilibrium points, and stability. Moreover, chaos synchronization implementation is also presented. A nonlinear control scheme was proposed to find the stability of error dynamics, which reduced the computational complexity of the synchronization algorithm. In comparison to the existing synchronization approaches, the controller in this paper was designed based on the analytical technique (linearization method), which does not require an auxiliary function (Lyapunov function) as in traditional methods. Numerical simulations were carried out by using MATLAB to validate the effectiveness of the analytical technique.

In this paper, we have discussed various topological properties of (Hausdörff) topological ternary semigroup and topological ternary group. We have proved that the Cartesian product of an arbitrary family of topological ternary semigroups is again a topological ternary semigroup. We have investigated the existence of identity and idempotent in a topological ternary semigroup and discussed a method to topologize a ternary semigroup (group) with a compatible topology using some family of pseudometrics. Finally, we have proved that a compact topological ternary semigroup contains a ternary subgroup.

The first multiplicative Zagreb index ${\mathrm{\Pi }}_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index ${\mathrm{\Pi }}_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for ${\mathrm{\Pi }}_1$ and upper bound for ${\mathrm{\Pi }}_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds.