Journal of Mathematical Extension
Volume:15 Issue: 1, Winter 2021

Let R be a commutative ring with identity and n a positive integer greater than 1. In this paper, we introduce the concept of semi-n-absorbing submodules. A proper submodule N of an R-module M is called a semi-n-absorbing submodule of M if whenever a \in R, x \in M and a^nx \in N then ax \in N ora^n \in (N :R M). A number of results concerning semi-n-absorbing submodulesand examples of them are given. It is shown that if N is a semi-n-absorbing submodule of M and F is a flat R-module such that F \otimes N is proper in F \otimes M then F \otimes N is semi-n-absorbing in F \otimes M. We show that the converse holds when F is faithfully flat.

The best proximity pair problem in Banach spaces has already been examined by considering some special conditions.In this article, we mostly pay attention to the existence and uniqueness of the best proximity pair for T-absolutely directsets. This investigation is based on some interesting relations existing in Banach lattices, in which T : A ! B is an arbitrarymap.

Let X be a real 2-Banach space. We follow Gunawan, Mashadi, Gemawati, Nursupiamin and Siwaningrum in saying that x is orthogonal to y if there exists a subspace V of X with codim(V ) = 1 such that ∥x+λy; z∥ ≥ ∥x; z∥ for every z ∈ V and λ ∈ R. In this paper we prove that every linear self mapping T: X → X which preserve orthogonality is a 2-isometry multiplied by a constant.

Let G be a digraph with positive edge weights as well as s and t be two vertices of G. The marking problem (MP) states how to mark some edges of G with T and F, where every path starting at source s will reach target t and the total weight of the marked edges is minimal. When traversing the digraph, T-marked edges should be followed while Fmarked edges should not. The basic applications and properties of the marking problem have been investigated in [1]. This paper provides new contributions to the marking problem as follows: (i) the MP is NP-Complete even if the underlying digraph is an unweighted binary DAG; (ii) the MP cannot be approximated within α logn in an unweighted DAG with n vertices and even in an unweighted binary DAG. Furthermore, a lower bound to the optimal solution of the MP is provided. We also study the complexity and challenges of the marking problem in program flow graphs.

Rail transportation plays an important role in one country's economic development. Therefore, many researchers have focused attention on the measurement of efficiency and ranking in the rail transportation industry. Evaluating organizational performance can produce significant effects on the organization’s activities. Data Envelopment Analysis (DEA) is a mathematical programming method, which measures the relative efficiency of the organizational units, which have multiple inputs and outputs. In this paper, we proposed a method for ranking all the Decision Making Units (DMUs) that is based on strong and weak supporting hyperplanes. As an applied project, we evaluate rail freight and passenger transportation in some Asian countries. The ranking is based on data from the International Union of Railways (UIC) in 2016.

‎The concept of limit summability of real functions was introduced and studied by the second author in 2001‎. ‎Concerning this‎, ‎the limit summand function $f_{\sigma}$ corresponds to a real or complex function $f$ with domain $D_f\supseteq\mathbb{N^*}$‎, ‎which satisfies the difference functional equation $f_{\sigma}(x) = f(x)‎ + ‎f_{\sigma}(x-1)$‎. ‎By considering the special case of $g:\mathbb{R^+} \rightarrow \mathbb{R^+}$ and by putting $f=\log g$‎, ‎he showed that $\Gamma$-type functions can be considered as a special case of limit summability and the basic relation $g^*(x+1) = e^{f_{\sigma}(x)}$ holds (where $g^*$ is its $\Gamma$-type function)‎. ‎As a result‎, ‎he improved one of the main theorems for $\Gamma$-type functions due to Webster in 1997‎. ‎Regarding the Gauss multiplication formula for $\Gamma$-type functions‎, ‎we introduce its dual for limit summand functions‎, ‎that is, Gauss \textit{summation} formula‎. ‎Also‎, ‎we show that not only the Gauss multiplication for $\Gamma$-type functions is a simple result of this formula‎, ‎but also provide an improvement of it with several consequences and applications‎. ‎Finally‎,‎‎ ‎we point out that in two theorems of Webster some conditions are superfluous‎.

In this work, we present and study new ordered separation axioms,namely supra Tci-ordered spaces (briey, STci-ordered spaces), for i = 0; 12 ; 1; 112 ; 2.We illustrate the relationships among these ordered spaces with the help of examplesand we point out under what conditions the initiated ordered separationaxioms are hereditary properties. Also, we derive some results which connect someof STci-ordered spaces with some topological notions such as supra limit points andsupra disconnected spaces, and with some algebra notions such as largest and smallestelements. Furthermore, we investigate the image of theses ordered spaces underS?-homeomorphism maps. Finally, we verify that the nite ordered product of STci -ordered spaces is STci-ordered, for i = 0; 12 ; 1; 112 .

Let $\Gamma=(G,\sigma)$ be a signed graph, where $G$ is the underlying simple graph and $\sigma : E(G) \longrightarrow \lbrace -,+\rbrace$ is the sign function on the edges of $G$. The adjacency matrix of a signed graph has $-1$ or $+1$ for adjacent vertices, depending on the sign of the connecting edges. Let $\Gamma=(K_{n},\bigcup_{i=1}^{m}P_{r_i}^- )$ be a signed complete graph whose negative edges induce a subgraph which is the disjoint union of $ m$ distinct paths. In this paper, by a constructive method, we obtain $n-1+\Sigma _{i=1}^{m}\big(\lfloor \frac{r_i}{2}\rfloor-r_i\big) $ eigenvalues of $\Gamma$, where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

‎Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$‎. ‎It is well-known that $G/H$ as a homogeneous space admits a strongly quasi invariant measure and the linear mapping $T_H$ of $L^1(G)$ into $L^1(G/H)$ is bounded and surjective‎. ‎In this note it is shown that by means of complex interpolation theorem‎, ‎that under restrictions on weight function $\omega$‎, ‎the mapping $T_H$ of weighted spaces $L^p(G,\omega)$ into $L^p(G/H,\varpi)$ is well-defined‎, ‎bounded linear and surjective‎ , ‎for $1\leq p \leq \infty$‎.

In this paper, we first obtain two weighted identities for twice partially differentiablemappings. Moreover, utilizing these equalities, we establish the weighted Herrmite-Hadamard typeinequalities and weighted Simpson type inequalities for co-ordinated convex functions in a rectanglefrom the plane R2, respectivelly. The results given in this paper provide generalizations of someresult established in earlier works.

Data envelopment analysis (DEA) is used as a handy tool to evaluate a set of decision-making units (DMUs) forming a sustainable supply chain (SSC). Furthermore, the Malmquist Productivity Index (MPI) is used to determine the level of progress or regression in the DMUs. In this paper, we calculate the MPI in a sustainable supply chain using non-radial DEA models. Generally speaking, we are after the sustainability of economic, environmental and social conditions in a supply chain, and the subject of sustainable supply chain is quite important in this respect. Thus, in this paper, MPI is calculated in a sustainable supply chain using the non-radial Enhanced Russell Measure (ERM). Moreover, our applied study involves an evaluation of sustainable supply chains in Iranian oil refineries.

‎In this paper‎, ‎we investigate the existence of a solution for a pointwise defined‎, ‎multi-singular fractional differential equation in some regions with some boundary conditions‎.