Construction of the Rough Quotient Modules over the Rough Ring by Using Coset Concepts

Given an ordered pair $(U, \theta)$ where $U$ is the set universe and $\theta$ is an equivalence relation on the set $U$ is called an approximation space. The equivalence relation $\theta$ is a relation that is reflexive, symmetric, and transitive. If the set $X \subseteq U$, then we can determine the upper approximation of the set $X$, denoted by $\overline{Apr}(X)$, and the lower approximation of the set $X$, denoted by $\underline{Apr}(X)$. The set $X$ is said to be a rough set on $(U, \theta)$ if and only if $\overline{Apr}(X)-\underline{Apr}(X) \neq \emptyset$. A rough set $X$ is a rough module if it satisfies certain axioms. This paper discusses the construction of a rough quotient module over a rough ring using the coset concept to determine its equivalence classes and discusses the properties of a rough quotient module over a rough ring related to a rough torsion module

BOUNDING LINEAR-WIDTH AND DISTANCE-WIDTH USING FEEDBACK VERTEX SET AND MM-WIDTH FOR GRAPH

Studying the upper and lower bounds of graph parameters is crucial for understanding the complexity and tractability of computational problems, optimizing algorithms, and revealing structural properties of various graph classes. In this brief paper, we explore the upper and lower bounds of graph parameters, including path-distance-width, MM-Width, Feedback Vertex Set, and linear-width. These bounds are crucial for understanding the complexity and structure of graphs

COORDINATING AND OPTIMIZING TWO-WAREHOUSE INVENTORY SYSTEMS: A MATHEMATICAL PROGRAMMING APPROACH

Effective supplier and carrier selection plays a pivotal role in supply chain management, ensuring maximum profitability. This study introduces an innovative decision-support system designed for supplier and carrier selection problems in static two-warehouse inventory systems. The model assumes warehouse collaboration, where warehouses consolidate efforts to fulfill overall demand. To address this, a mathematical programming approach is developed and solved using the LINGO 21.0 optimization software. Experimental results reveal that the proposed model delivers optimal decisions. Even though challenges are still available on the constraint functions and the derivation of parameters' values, the results provide positive managerial insights that offer valuable tools for stakeholders to improve supply chain efficiency

An Algorithm for Generalized Conversion to Normal Distribution for Independent and Identically Distributed Random Variables

The paper analyzes an efficient alternative to the Box-Cox and Johnson’s transformation to normality methods which operates under fairly general settings. The method hinges on two results in mathematical statistics: the fact that the cumulative distribution function F(x) of a random variable x always has a U(0,1) distribution and the Box-Mueller transformation of uniform random variables to standard normal random variables.  Bounds for the Kolmogorov-Smirnov statistic between the distribution of the transformed observations and the normal distribution are provided by numerical simulation and by appealing to the Dvoretzky-Kiefer- Wolfowitz inequality

THE L(2,1)-LABELING OF MONGOLIAN TENT, LOBSTER, TRIANGULAR SNAKE, AND KAYAK PADDLE GRAPH

Let G = (V,E) be a simple graph. L(2, 1)−labeling defined as a functionf : V (G) → N0 such that, x and y are two adjacent vertices in V, then if x andy are adjacent to each other, |f(y) − f(x)| ≥ 2 and if x and y have the distance 2,|f(y) − f(x)| ≥ 1. The L(2, 1)-labeling number of G, called λ2,1(G), is the smallestnumbermof G. In this paper, we will further discuss the L(2, 1)-labeling of mongoliantent, lobster, triangular snake, and kayak paddle.Keywords: L(2,1)-Labeling, mongolian tent, lobster, triangular snake, kayak paddle.

ALGEBRAIC STRUCTURES IN HEREDITY HUMAN BLOOD GROUP SYSTEM

Marriage or in this case the researcher calls it "cross-operation" between two individuals (male and female) who have the same or different blood type has the probability to produce children (offspring) with the same blood type as one of the parents or even have a completely different blood type with both of them, whether it is the ABO blood type system or MN if it is associated with the rhesus system or not. The cross-operation between two individuals can be viewed from a mathematical perspective as an algebraic structure with one closed binary operation (OB). The cross-operation of ABO blood group system is an algebraic structure in groupoid form. The cross-operation of MN blood group system is an algebraic structure in groupoid form. And finally, the cross-operation of ABO and MN blood group systems when associated with the rhesus blood group system is an algebraic structure in groupoid form

MIXTURE PURIFICATION MODEL WITH CASCADING TANK CONFIGURATION

Consider mixing problems which are often found in Calculus or Differential Equation courses. Under some assumptions, this problem can be used to model the purification process in a polluted mixture. In this case, the cascading configuration will be investigated for modelling the spread of pollution from one mixture to another. There are two main problems: finding time needed so the amount of pollutant in mixture inside the certain tank does not exceed certain threshold and finding the number of tanks needed so that the amount of mixture in the last tank does not exceed certain threshold. The solution for the second problem will be simplified by using Stirling approximation, which approximates factorial into exponential term. For the first problem, the time needed depends on the number of tanks, initial value of the pollutant, the rate of flow, and the volume of solution inside the tanks. For the second problem, the number of tanks only depends on the initial value of the pollutant

SOME PROPERTIES OF ALMOST JOINTLY PRIME (R,S)-SUBMODULES

Let and be rings with identity. The definition of prime submodule has been generalized to the almost prime submodule. In addition, the definition of prime submodule has also been carried over to the (,)-module structure, which is called jointly prime (,)-submodules. However, as a generalization of prime submodules, the concept of almost prime submodules has not been carried over to (,)-module structures. In this paper, we construct the definition of almost jointly prime (,)-submodules as the generalization of jointly prime (,)-submodules. We also present several necessary and sufficient conditions for an (,)-submodule to be an almost jointly prime (,)-submodule

ANALYSIS OF A NON LINEAR DYNAMICS MODEL FOR TRANSMISSION TUBERCULOSIS IN NIGERIA INCORPORATING TREATMENT AND VACCINATION

This work models and analyzes the transmission of tuberculosis infection with the impact of vaccination and treatment on the bacteria in Nigeria from 2010 to 2022 incorporating treatment and vaccination. The susceptible-vaccinated-Exposed-Infected-Recovered (SVEIR) model is used for the transmission of the bacteria in which the with immigrants are exposed to infection infectious individuals, and it is assumed that there is permanent immunity and homogenous mixing against the bacteria. The constant immigration of the infected individuals into the population makes it impossible for the disease to die out and so there is no disease-free equilibrium. The fraction of chemoprophylaxis Bacillus Calmette-Guerin (BCG) was incorporated into the model equation for successful vaccination. Stability analysis shows that a disease free equilibrum is locally asymptotically stable for R01 which can wipe out the whole population. Hence, treatment and vaccination are the measures that can reduce below 1 in order to control tuberculosis

BOUNDED TREE-DEPTH, PATH-DISTANCE-WIDTH, AND LINEAR-WIDTH OF GRAPHS

The study of width parameters and related graph parameters is an activearea of research in graph theory. In this brief paper, we explore the upper and lowerbounds of graph parameters, including path-distance-width, tree-distance-width, tree-depth, and linear-width. These bounds are crucial for understanding the complexityand structure of graphs