Asian-European Journal of Mathematics

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that ${𝔽}_q$ is the finite field of order $q,$ where $q$ is a power of the prime $p,$ and $p,\ell$ are distinct odd primes, and $m,n$ are positive integers. We determine generator polynomials of all constacyclic codes of length $8{\ell }^m{p}^n$ over the finite field ${𝔽}_q.$ We also determine their dual codes.

In this paper, we investigate the two-component Camassa–Holm equation with fractional dissipation

Some properties of the weak product of graphs on lattices are studied. We consider the weak product of the atom-based graphs, the incomparability graphs and the zero-divisor graphs of lattices. The number of edges in the weak product of such graphs is found. It is shown that the weak direct product of the atom-based graphs of two lattices $L_1$ and $L_2$ cannot be an induced subgraph of the atom-based graph of $L_1\times L_2$.

The focus of this paper is to study the two-variable Kauffman polynomials $L$ and $F$, and the one-variable BLM/Ho polynomial $Q$ of $(2,n)$-torus link as the Fibonacci-type polynomials and to express the Kauffman polynomials in terms of the BLM/Ho polynomial. For this purpose, we prove that each of the examined polynomials of $(2,n)$-torus link can be determined by a third-order recurrence relation and give the recursive properties of them. We correlate these polynomials with the Fibonacci-type polynomials. By using the relations between the BLM/Ho polynomials and Fibonacci-type polynomials, we express the Kauffman polynomials in terms of the BLM/Ho polynomials.

This paper presents biorthogonal wavelet-based multigrid (BWMG) and full approximation scheme (FAS) for the numerical solution of parabolic partial differential equations (PPDEs), which are working horse behind many commercial applications like finger print image compression. Performance of the proposed schemes is better than the existing ones in terms of super convergence with less computational time. Some of the test problems are taken to demonstrate the applicability and validity of the method.

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

Let $T(X)$ be a semigroup of all transformations on a set $X.$ Let $Y$ be a fixed nonempty subset of $X$ and $S(X,Y)=\{f\in T(X)|(Y)f\subseteq Y\}.$ Then $S(X,Y)$ is a subsemigroup of $T(X)$ which leaves a subset $Y$ of $X$ invariant. If $Y=X,$ then $S(X,Y)=S(X,X)=T(X).$ In this paper, we give necessary and sufficient conditions for elements in $S(X,Y)$ to be left or right magnifying.

Let $\overline{\mathrm{\Gamma }(R)}$ be the complement of the zero-divisor graph of a finite commutative ring $R$. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that $\overline{\mathrm{\Gamma }(R)}$ is a divisor graph if $R$ is a local ring. It is shown that when $R$ is a product of two local rings, then $\overline{\mathrm{\Gamma }(R)}$ is a divisor graph if one of them is an integral domain. Further, if $|Ass(R)|=2$, then $\overline{\mathrm{\Gamma }(R)}$ is a divisor graph.

Let $T(X)$ be the semigroup of all transformations on a set $X$. For an arbitrary equivalence relation $E$ on $X$ and a cross-section $R$ of the partition $X/E$ induced by $E$, let

In this paper, a condition which implies the continuity of an $h$-convex function is investigated. In fact, any $h$-convex function bounded from above is continuous if the function $h$ satisfies a certain condition which is called pre-continuity condition.

For positive integer $k$, a graph $G$ is said to be $k$-magic if the edges of $G$ can be labeled with the nonzero elements of Abelian group ${\mathbb{Z}}_k$, where ${\mathbb{Z}}_1=\mathbb{Z}$ (the set of integers) and ${\mathbb{Z}}_k$ is the group of integers mod $k\ge 2$, so that the sum of the labels of the edges incident to any vertex of $G$ is the same. When this constant sum is $0$, we say that $G$ is a zero-sum $k$-magic graph. The set of all $k$ for which $G$ is a zero-sum $k$-magic graph is the null set of $G$. In this paper, we will completely determine the null set of the join of a finite number of paths.

The main focus of the paper is to study the performance of Arithmetic Mean (AM) iterative method in solving large and nonsingular dense linear system arising from the composite 6-point closed Newton–Cotes quadrature (6-CCNC) approximation of inhomogeneous linear Fredholm integral equations of the second kind. The derivation and implementation of the method are discussed. Some numerical results are also presented.

Let $V$ be a vector space and $T(V)$ denote the semigroup (under the composition of maps) of all linear transformations from $V$ into itself. For a fixed subspace $W$ of $V$, let $\mathrm{\text{Fix}}(V,W)$ be the subsemigroup of $T(V)$ consisting of all linear transformations on $V$ which fix all elements in $W$. In this paper, we describe Green’s relations, regularity and ideals of $\mathrm{\text{Fix}}(V,W)$; and characterize when $\mathrm{\text{Fix}}(V,W)$ is factorizable, unit-regular and directly finite, from which the results on $T(V)$ can be recaptured easily when taking $W$ as a zero subspace of $V$.

In this paper, we obtain new inequalities for relative operator entropy $S(A|B)$ in terms of Tsallis’ relative entropy $T_{\pm t}(A|B),$ $t>0$ in the case of positive invertible operators $A,$ $B$. Further bounds for $B\ge A$ are also provided.

Let $\rho$ and $\sigma$ be two central relations on a finite set $A$. It is known from Rosenberg’s classification theorem (1965) that the clones $\mathrm{\text{Pol}}\rho$ and $\mathrm{\text{Pol}}\sigma$ which consist of all operations on $A$ that preserve $\rho$, respectively, $\sigma$ are among the maximal clones on $A$. In this paper, we find all central relations $\sigma$ such that the clone $\mathrm{\text{Pol}}\{\rho ,\sigma \}$ is a maximal subclone of $\mathrm{\text{Pol}}\rho$, where $\rho$ is a fixed central relation.

We establish analogues of Heisenberg uncertainty inequality for some classes of Lie groups, such as connected and simply connected nilpotent Lie groups, diamond Lie groups and Heisenberg motion groups.

In this paper, we establish the structure of $U(F{D}_7)$, $U(F{D}_{12})$, $U(F{D}_{14})$ and $U(F{D}_{24})$ where $F$ is a finite field and $D_n$ is the dihedral group of order $2n$. For an odd prime $p$, we also establish the structure of $U(F{D}_{2^k\cdot p})$, when characteristic of $F$ is $2$ or $p$.

We provide a new representation of an $ℝ$-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the $ℝ$-trees with radial and river metrics, and these sets of metric rays. In stochastic analysis, these graphical representation theorems are of particular interest in identifying Brownian motions indexed by $ℝ$-trees.