Asian-European Journal of Mathematics

Metric dimension and local metric dimension are distance-based tools that find their applications in robot navigation, combinatorial optimization, pattern recognition, image processing, integer programming, and many more. A study based on the selection of the minimum number of vertices in a graph that will separate all the neighboring vertices in terms of distance between the vertices is given by the local metric basis problem. It is well known that the problem of finding the local metric basis of a graph is NP-complete. In this paper, the local metric dimension of double wheel graph ${\mathrm{\text{DW}}}_n$, generalized wheel graph $W_{n,m}$, and closed helm graph ${\mathrm{\text{CH}}}_n$ had been computed. It has been found that these graphs have unbounded local metric dimensions.

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I+J$ is a prime ideal of $R$. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus $2$.

Let $R$ be a commutative ring with non-zero identity, $S\subseteq R$ a multiplicatively closed subset of $R$ and $M$ a unital $R$-module. In this paper, we introduce the concepts of $S$-almost semiprime submodules and $S$-weakly semiprime submodules as a generalization of $S$-semiprime submodules. We investigate some basic properties of $S$-almost semiprime and $S$-weakly semiprime submodules and give some characterizations of them, especially, for (finitely generated faithful) multiplication modules.

In this paper, some generalizations of lower bound estimates for the maximum modulus of the $t$th polar derivative $D_{{\xi }_t}{D}_{{\xi }_{t-1}}\dots D_{{\xi }_2}{D}_{{\xi }_1}$, where ${\xi }_t\in ℂ$, $1\le t<n$ of polynomials are established under the assumption that one of the underlying polynomials $p$ and $q$ does not vanish in $\left|z\right|>k,k\le 1$. The obtained estimates include several known lower bounds for polar derivatives as special cases.

In this work, we define a new notion of $\mu F$-contraction in Banach spaces, by modeling with measure of non-compactness the well-known concept of $F$-contraction given by Wardowski in metric spaces. We prove a fixed-point theorem and a common fixed-point theorem for our contraction. Our results extend the well-known Darbo contraction principle and large contractions in the literature. Furthermore, the paper contains examples and applications that show the validity of our results.

In this paper, we study the class of Baire one path systems as a metric space and as a multifunction and show that some known classes of path systems such as continuous path systems, composite continuous path systems, and first return path systems are subclasses of the class of Baire one path systems. In particular, we show that a path system is composite continuous if and only if it is Baire${}^{\ast }$ one path system. We also show that for a composite continuous path system $E=\{E_x:x\in [0,1]\}$, the upper (respectively, lower) extreme path derivatives $\overline{f_E^{\prime }}$ (respectively, $\underline{f_E^{\prime }}$) of a function $f\in B_1$ are upper (respectively, lower) Baire two functions, and when $f$ is $E$-differentiable, $f_E^{\prime }\in B_2$.

Let a $2\pi$-periodic function $f\in ℂ$ change its monotonicity at a finitely even number of points $y_i$ of the period. The degree of approximation of this $f$ by trigonometric polynomials which are comonotone with it, i.e. that change their monotonicity exactly at the points $y_i$ where $f$ does, is restricted by ${\omega }_2(f,\pi /n)$ (with a constant depending on the location of these $y_i$). Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to $\pi /n$ about the points $y_i$ (so-called nearly comonotone approximation) allows the polynomials to achieve the approximation rate of ${\omega }_3$. By constructing a counterexample, we show that even with the relaxation of the comonotonicity requirement for the polynomials on sets with measures approaching $0$, ${\omega }_4$ is not reachable.

In this study, the concepts of lacunary $(H,1)-$summability of order $\alpha$, lacunary strongly harmonic summability of order $\alpha$, lacunary statistical logarithmic convergence of order $\alpha$ of sequences of real numbers are introduced and relations between these concepts are investigated.

Let ${𝒢}$ be a group, ${ℱ}$ be a finite field with a positive characteristic $p>0$, then ${ℱ}{𝒢}$ is the group algebra of a group ${𝒢}$ over ${ℱ}$ with a positive characteristic $p>0$. In this paper, the normal complement problem on semisimple group algebra ${ℱ}{𝒢}$ is examined. The normal complement problem of groups of order up to 16 in their corresponding unit groups ${𝒰}({ℱ}{𝒢})$ has already been discussed. We have proved that up to isomorphism all the three non-abelian groups of order 18 and up to isomorphism all three non-abelian groups of order 20 do not have normal complement in their corresponding unit groups ${𝒰}({ℱ}{𝒢})$.

By introducing matrices of linear bounded infinite-dimensional operators defined over some Banach spaces, and within the context of study, we make use of the definition of row strict diagonal dominance property to construct a generalization version of the Jacobi Under-Relaxation and the Successive Under-Relaxation iterative methods. The convergence analyses of the two new iterative methods are provided, and a numerical application to solve one Fredholm integral equation is presented to show the generalized methods’ effectiveness compared with their conventional opponents.

For fixed integers $1\le i<j\le n$ and $\sigma \in S_n$, we study the faithful representation of superassociative algebras by two kinds of functions of the arity $n$, the so called $(i,j)$-commutative and $\sigma$-commutative. The automorphism on the algebra of all $(i,j)$-commutative functions is determined. In general, an isomorphism from a superassociative system which consists of a family of nonempty sets and a family of operations satisfying the axiom of superassociativity to a system of full multiplace functions is discussed. Particularly, we also provide necessary and sufficient conditions under which a superassociative system and a system of $\sigma$-commutative functions are isomorphic. With these results, a close connection with the theory of terms and tree languages is described.

This study aims to provide a non-limiting factor auxiliary function in a general form. This function, under some assumptions, is proven to fulfill the properties of filling functions. The properties of the proposed method, which include continuously differentiable and the absence of any limiting factor, provide an advantage during numerical experiments for solving some global optimization problems. A numerical comparison was performed, and the results confirmed that our filling function has several advantages compared with other filling functions. Moreover, numerical computational results demonstrate the reliability and effectiveness of the proposed filling function.

The general rotational surfaces of ${𝔼}^4$ were first studied by Moore. The Vranceanu surfaces are special examples of this kind of surfaces. These constant-ratio surfaces are surfaces for which the ratio of the norms of the tangent and normal components of the position vector fields is constant. However, spherical surfaces and conical surfaces are also trivial examples of constant-ratio surfaces. Thus, if the norms of the tangent or normal components of the position vector fields are constant, then the given surface is called $T$-constant or $N$-constant, respectively. In this paper, we considered three types of rotational surfaces lying in $4$-dimensional Euclidean space ${𝔼}^4$. We have obtained the necessary and sufficient conditions for these surfaces to satisfy the $T$-constant, $N$-constant or constant-ratio conditions. With the help of these results, we characterized the meridian curves of the surfaces. Further, we also give some examples to support the results obtained.

The Generalized mixed equilibrium problems, which includes the equilibrium and mixed equilibrium problem of monotone type mapping is considered in this paper with the fixed point of Bregman strongly nonexpansive mapping in the framework of real reflexive Banach space. We approximate the common solution of the system of generalized mixed equilibrium and fixed point problems for the finite family of Bregman strongly nonexpansive mappings using an inertial Halpern method. Using our iterative method, we prove a strong convergence result for solving the aforementioned problems. We also present some applications and numerical examples to demonstrate the performance of our iterative method. Our result improves and extends some important results presented by authors in the literature.

In this paper, we extend the expansivity, pseudo trajectory tracing property and hyperbolicity of linear dynamical systems for the random view point. We show that to a random linear cocycle $A,$ it is expansive if and only if it has the generalized pseudo trajectory tracing property. Moreover, we show that $A$ is topologically stable if and only if it is structurally stable.

In this paper, we examine the relationship between a recent new discrete majorization-type inequality and classical majorization-type inequalities. The multiplicative analog of the studied new inequality is also given, which is a wide generalization of Weyl’s inequality. As an application, we give a parametric refinement of Popoviciu’s version of the Petrović inequality.

In this paper, we introduce the notions of product measurable convergence, deferred Cesàro statistical product measurable convergence and deferred Cesàro statistical Lebesgue product measurable convergence for sequences of measurable functions on product measure spaces. We establish some fundamental relations among these convergences and also give several explanatory examples in support of our definitions and results. Finally, as an application, we prove a new version of Korovkin-type approximation theorems for sequences of measurable functions on product measure spaces by using the notion of deferred Cesàro statistical Lebesgue product measurable convergence.

Let ${ℛ}$ be commutative ring and $Z^{\ast }({ℛ})$ be the set of all non-zero zero divisors of ${ℛ}$. Then $\mathrm{\Gamma }({ℛ})$ is said to be the zero divisor graph if and only if $a\cdot b=0$ where $a,b\in V(\mathrm{\Gamma }({ℛ}))$ and $(a,b)\in E(\mathrm{\Gamma }({ℛ}))$. Graph energy ${ℰ}(\mathrm{\Gamma }({ℛ}))$ is defined as the sum of the absolute eigenvalues of the adjacency matrix ${ℳ}(\mathrm{\Gamma }({ℛ}))$, then ${ℰ}(\mathrm{\Gamma }({ℛ}))={\sum }_{i=1}^n\left|{\lambda }_i\right|$. Wiener index $W(\mathrm{\Gamma }({ℛ}))$ is defined as the sum of all distance between pairs of vertices $a$ and $b$, then $W(\mathrm{\Gamma }({ℛ}))={\sum }_{a,b\in V(\mathrm{\Gamma }({ℛ}))}d(a,b)$. In this paper, we compute the graph energy from the adjacency matrix of the zero divisor graph and the Wiener index from the zero divisor graph associated with commutative rings.

In this paper, we consider a multi-term fractional differential equation involving the $\psi$-Caputo fractional derivative. Under some suitable hypotheses, we prove that our proposed problem has at least one mild solution expressed in terms of the $\psi$-$(\gamma ,\sigma )$-resolvent. As an application, we provide a suitable example at the end of this work to illustrate our theoretical results.

In this paper, we consider the unit graph $G({\mathbb{Z}}_n)$, where $n=p_1^{n_1}\text{ or }{p}_1^{n_1}{p}_2^{n_2}\text{ or }{p}_1^{n_1}{p}_2^{n_2}{p}_3^{n_3}$ and $p_1,p_2,p_3$ are distinct primes. For any prime $q$, we construct $q$-ary linear codes from the incidence matrix of the unit graph $G({\mathbb{Z}}_n)$ with their parameters. We also prove that the dual of the constructed codes have minimum distance either three or four. Lastly, we stated two conjectures on diameter of unit graph $G({\mathbb{Z}}_n)$ and linear codes constructed from the incidence matrix of the unit graph $G({\mathbb{Z}}_n)$ for any integer $n$.