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

HOW TO COMBINE VAM AND DIJKSTRA’S ALGORITHM

Solving transportation problems sometimes does not only require using one method or algorithm. Sometimes it is necessary to use several methods or algorithms at once. In this research, combining the Vogel’s Approximation Method (VAM) and Dijkstra algorithm can be carried out if three assumptions are met. These three assumptions are based on the characteristics of each VAM and Dijkstra’s algorithm, as well as the compatibility between the two

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

HIERARCHICAL BAYESIAN SMALL AREA ESTIMATION ON OVERDISPERSED DATA: WORKERS WITH DISABILITIES IN INDONESIA

Persons with disabilities encounterdifficulties in accessing essentialservices, including employment, healthcare, information, and political participation. In line with the target 8.5 of the SDGs, efforts have been made to promotefull, productive, and decent employment for all, including for persons with disabilities. However, the majority ofworkers with disabilities in Indonesia remain concentrated in the informal sector during the period of 2022–2023. Unfortunately, data on workers with disabilities is currently only available at the national level. This limitation arises because the sample size of workers with disabilities is insufficient to meet the minimum requirements for direct estimation at the provincial level. Therefore, a Small Area Estimation approach is necessary to assess the participationof persons with disabilities in the workforce at more granular level, such as provinces. In this study, auxiliary variables such as the sex ratio, the number of residents who are shackled, and the availability of computer skills infrastructure were incorporated to the Small Area Estimation (SAE) framework. The Hierarchical Bayesian Poisson-Gamma was employed to improve the precision of direct estimation. The research results show that the HB Poisson-gamma estimator has better precision compared to the direct estimator

On the necessary and sufficient condition of a k-Euler pair

In this paper, we discuss George Andrews’ definition of an Euler pair andSubbarao’s generalization of the Euler pair to a k-Euler pair. Let N and M be non-empty sets of natural numbers. A pair (N, M) is called a k-Euler pair if, for any natural number n, the number of partitions of n into parts from N is equal to the number of partitions of n into parts  from M, with  the  condition  that  each  part  appears  fewer than k times. We further explore several theorems concerning Euler pairs that were established by Andrews and Subbarao, and we present proofs using a method distinct from those previously utilized

PMC-Labeling of Certain Classes of Graphs

In this paper, we investigate the PMC-labeling behavior of some new graphs such as the double fan graph, triple fan graph, $m$--enriched fan graph,  C_{n}--snake, stripe blade graph, G_{n}, Sf_{n} + K_{1}, armed helm graph, alternate armed helm graph and spectrum graph

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

EPIDEMIC ANALYSIS, MATHEMATICAL MODELLING AND NUMERICAL SIMULATION OF COVID-19 TRANSMISSION

This research develops a model with seven compartments SEIQDHR for the spread of COVID-19, with detected and treated individual behavior changes affecting disease transmission. The Next Generation Matrix is used to analyze local and global stability and to calculate the basic reproduction number. Then, the analysis of disease-free equilibrium and endemic equilibrium. Stability analysis shows that the equilibrium point is locally asymptotically stable when the basic reproduction number is less than one and globally asymptotically stable when it is greater than one. The results of the sensitivity analysis show that the transmission rate, the progression rate from exposure, and the detection rate are parameters that significantly influence the dynamics of disease spread. Numerical simulations were used to validate the analysis results and identify key parameters that contribute most to the spread of the disease among affected, infected, quarantined, diagnosed, and hospitalized individuals

Sharper Upper Bounds for Roots of Polynomials Generated by Positive Sequences

Finding sharp and easily computable upper bounds for the moduli of the roots of polynomials with real coefficients is a long-standing problem with applications in numerical analysis, control theory, and the study of linear recurrence relations. The classical bounds of Cauchy and Lagrange, despite their age, remain the most frequently used estimates because of their extreme simplicity. This paper introduces a new family of upper bounds specifically designed for polynomials whose coefficients are the initial terms of a positive real sequence a_n that does not grow too rapidly. For each such polynomial we construct an explicit number by taking the two largest values appearing among the (i+1)-th roots of the successive absolute differences of the sequence together with the simple quantity a_1+1, and adding them. We prove that the resulting value rigorously bounds the modulus of every root. A companion bound based on second differences is obtained as an immediate corollary. Extensive numerical tests on constant, arithmetic, harmonic, and exponential sequences show that the new estimates are often several times tighter than Cauchy’s bound and, in many cases, also outperform recently published refinements. The contribution is twofold: (i) a new, fully explicit bound using first differences, and (ii) an even sharper variant using second differences presented as a corollary