Asian-European Journal of Mathematics

Simply reducible groups are closely related to the eigenvalue problems in quantum theory and molecular symmetry in chemistry. Classification of simply reducible groups is still an open problem which is interesting to physicists. Since there are not many examples of simply reducible groups in literature at the moment, we try to find some examples of simply reducible groups as candidates for the classification. By studying the automorphism and inner automorphism groups of symmetric groups, dihedral groups, Clifford groups and Coxeter groups, we find some new examples of candidates. We use the computer algebra system GAP to get most of these automorphism and inner automorphism groups.

In this paper, we examine a novel method called the residual power series method (RPSM) which is used in finding an analytic approximate solution to the nonlinear temporal foam drainage equation of fractional order. The solution is obtained in a uniformly convergent series form. Also, 3D graphs for the solution are provided for different values of $\alpha$ which proves that the method is an effective and straightforward approach of providing excellent results for similar problems.

Let ${𝒫}(G)$ denote the power graph of a finite group $G$. Let $\rho (Q({𝒫}({\mathbb{Z}}_n)))$ denote the Signless Laplacian spectral radius of ${𝒫}({\mathbb{Z}}_n)$. In this paper, we give lower and upper bounds on $\rho (Q({𝒫}({\mathbb{Z}}_n)))$ for any $n>2$ and find those graphs for which the extremal values are attained. We give a comparison between the bounds obtained and exact value of $\rho (Q({𝒫}({\mathbb{Z}}_n)))$ for any $n>2$. We then find the eigenvalues of $Q({𝒫}(D_n))$ and give lower and upper bounds on the spectral radius of $Q({𝒫}(D_n))$. When $n=p^m$ and $n=pq,$ where $p,q$ are primes and $m$ is a positive integer, we obtain sharper bounds on $\rho (Q({𝒫}(D_n)))$. Finally, we make a conjecture on $\rho (Q({𝒫}(D_n)))$ for any $n>2$.

In this paper, we investigate the structure of a finite group $G=HK$ which is totally, mutually or mutually $m$-permutable product of proper subgroups $H$ and $K$ under theassumption that all maximal subgroups of $G$ are $GS$-groups.

The superposition operation $S^{n,A},$ $n\ge 1,$ $n\in \mathbb{N}$, maps to each $(n+1)$-tuple of $n$-ary operations on a set $A$ an $n$-ary operation on $A$ and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank $n$. A partial algebra of type $(n+1)$ which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank $n$. As a generalization of linear terms we define $r$-terms as terms where each variable occurs at most $r$-times. It will be proved that $n$-ary $r$-terms form partial Menger algebras of rank $n$. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider $r$-hypersubstitutions.

In this paper, we define and study the class $\gamma -B_{\sigma }(\alpha ,\beta )$ of generalized gamma-Bazilevic functions. Our main focus is to discuss certain problems such as inclusion results, covering theorem and radius problem.

A closed knight’s tour of a normal two-dimensional chessboard by using legal moves of the knight has been generalized in several ways. One way is to consider a closed knight’s tour on a ringboard of width $r$, which is the $m\times n$ chessboard with the middle part missing and the rim contains $r$ rows and $r$ columns. Another way is to stack $k$ copies of the $m\times n$ chessboard to construct an $m\times n\times k$ rectangular chessboard and the closed knight’s tour can be on the surface or within the $m\times n\times k$ rectangular chessboard. This paper combines these two ideas by stacking $k$ copies of $m\times n$ ringboard of width $r$, which we call the $(m,n,k,r)$-rectangular tube. We explore the existence and the nonexistence of closed knight’s tours for $(3,n,k,1)$-tube and $(m,m,k,1)$-tube.

In this paper, we introduce Padovan difference sequence spaces of fractional-order $\alpha ,$ ${\ell }_p({\tilde{P}}^{(\alpha )})$ $(1\le p\le \infty )$ by the composition of the fractional-order difference operator ${\mathrm{\Delta }}^{(\alpha )}$ and the Padovan matrix $\tilde{P}=({\tilde{p}}_{nk})$ defined by ${\mathrm{\Delta }}_k^{(\alpha )}(x)={\sum }_{i=0}^{\infty }{(-1)}^i\frac{\mathrm{\Gamma }(\alpha +1)}{i!\mathrm{\Gamma }(\alpha -i+1)}{x}_{k-i}$ and

respectively, where the sequence $({\tilde{p}}_k)$ is the Padovan sequence. We give some topological properties, Schauder basis and $\alpha$-, $\beta$- and $\gamma$-duals of the newly defined spaces. We characterize certain matrix classes related to the ${\ell }_p({\tilde{P}}^{(\alpha )})$ space. Finally, we characterize certain classes of compact operators on ${\ell }_p({\tilde{P}}^{(\alpha )})$ using Hausdorff measure of noncompactness.

Let us consider a mapping $\varphi :E(G)\to \{0,1,\dots ,k-1\}$ of a graph $G$, where $k$ is an integer, $2\le k\le |E(G)|$. The mapping $\varphi$ induces for every vertex $v$ of $G$ the label ${\varphi }^{\ast }(v)={\prod }_{uv\in E(G)}\varphi (uv)(modk)$. Let $e_{\varphi }(i)$ ($v_{\varphi }(i)$) denote the number of edges (vertices) in $G$ that are labeled with the number $i$ under the labeling $\varphi$, $0\le i\le k-1$.

The function $\varphi$ is called a $k$-total edge product cordial labeling of $G$ if $|(e_{\varphi }(i)+v_{\varphi }(i))-(e_{\varphi }(j)+v_{\varphi }(j))|\le 1$ for $0\le i<j\le k-1$. A graph $G$ with a $k$-total edge product cordial labeling is called a $k$-total edge product cordial graph.

In this paper, we prove that the grid graph $P_m\square P_n$ for $m,n\ge 2$ admits a $3$-total edge product cordial labeling.

In this paper, we investigate the uniqueness problems of meromorphic functions concerning differential polynomials sharing a small function. In particular, the resuts of the paper improve the results due to Waghamore and Anand [11].

In this paper, we consider an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-m})$ with $-m\equiv 3$ (mod $4$). In particular, we study the ring of integers corresponding to the field $K$ and visualize the form of ${𝜖}_K/p{𝜖}_K$. We also consider lattices over the ring of integers ${𝜖}_K$ and discuss the theta series to see its relation with the weight enumerator. As a consequence, we will see how the theta series differs for different $m$ and $m^{\prime }$.

We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal’s triangle. We define the Gauss Narayana numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we show that there is a relationship between the derivatives of the new Narayana polynomials and Pascal’s triangle. We also explain the relationship between the new Narayana polynomials and the known Pell numbers. Finally, we give the Hankel transform of the new Narayana polynomials.

In this paper, we define a new family of Gauss $k$-Lucas numbers. We give the relationships between the family of the Gauss $k$-Lucas numbers and the known Gauss Lucas numbers. We also define the generalized polynomials for these numbers. We obtain some interesting properties of the polynomials. We also give the relationships between the generalized Gauss $k$-Lucas polynomials and the known Gauss Lucas polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we prove Cassini’s identities for the families and their polynomials.

In this paper, we introduce a new class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.

In this paper, Lie super-bialgebra structures on the $N=2$ super-BMS₃ algebra are investigated. By computing derivations from the super-BMS₃ to the tensor product of its adjoint module, we obtain that all Lie bialgebra structures on the $N=2$ super-BMS₃ Algebra are triangular coboundary.

In this paper, we introduce new classes of linear operators so called $C_0$-groups, $C$-groups and cosine families of bounded linear operators on non-archimedean Banach spaces over non-archimedean complete valued field $K$. We show some results about it.

Based on a theorem of McCoy on commutative rings, Nielsen called a ring $R$ right McCoy if for any nonzero polynomials $f(x),g(x)$ over $R$, $f(x)g(x)=0$ implies $f(x)r=0$ for some $0\ne r\in R$. In this note, we introduce and investigate McCoy and $CN$-properties of Hurwitz series ring $HR$ and its Hurwitz polynomial subring $hR$. We show that when $R$ is a reversible or duo ring and $\mathrm{\text{char}}(R)=0,$ then the Hurwitz polynomial ring $hR$ is McCoy.

The concept of a $\delta$-ideal is introduced in a pseudo-complemented distributive join semilattice with $0$ and some properties of these ideals are obtained. We also give a characterization for a pseudo-complemented distributive join-semilattice to be a Stone join-semilattice. Also, in a distributive join-semilattice a characterization for an ideal to be a $\delta$-ideal is proved.

In this paper, we consider all groups of order dividing $p^5$. We obtain the explicit structure of the non-abelian tensor square, non-abelian exterior square, tensor center, exterior center, the third homotopy group of suspension of an Eilenberg–MacLane space $k(G,1)$ and $▿(G)$ of such groups.