Research Updates in Mathematics and Computer Science Vol. 3

In this paper, we develop a finite capacity Multi-server Markovian queuing system with encourage arrival, defeat balking and feedback customers. The customer arrivals follow Poisson process with mean arrival rate \(\lambda\) and service time follow exponential distribution with parameter \(\mu\). The steady state solution and the system characteristic are derived for this model. The analytical results are numerically illustrated and the effects of the nodal parameters on the system characteristics are presented.

Methods of constructing the optimum chemical balance weighing designs from symmetric balanced incomplete block designs are proposed with illustration. As a by-product pairwise efficiency and variance balanced designs are also obtained. In this research, we have significantly shown that the obtained designs are pairwise balanced as well as efficiency balanced. The only limitation of this research is that the obtained pairwise balanced designs are all have large number of replications.

This study aimed to investigate the pH associated with micronutrients in soil samples. The characterization of the soil involved analyzing various factors, including pH, Sulfur, Zinc, Iron, Copper, Manganese, and Boron. A total of 500 soil samples were collected and categorized into four pH ranges: moderately acidic (5.1-6.0%), slightly acidic (6.1-6.5%), neutral (6.6-7.5%), and slightly alkaline (7.6-8.5%). The pH levels were neutral, indicating an ideal condition for maximum availability of primary nutrients essential for plant growth. The collected data was subjected to multinomial logistic regression and multivariate linear regression analyses. A formula was derived to determine the soil micronutrients based on the corresponding pH levels. The statistical tests of significance using linear regression indicated significant differences (P>0.05) between the pH values of the soil samples for Sulfur, Manganese, and Boron. Additionally, correlation analysis explored the relationships among different soil parameters. The likelihood ratio test supported a relationship between pH and micronutrient levels. The findings demonstrated that pH levels can be a predictive indicator of micronutrient performance in soil. Model evaluation, including the goodness of fit tests and pseudo-R-squares, accounted for 17% of the overall assessment. Moreover, the Multinomial logistic regression analysis achieved a classification accuracy of 64% for predicting pH levels. These findings have important implications for effective soil management strategies, aiding in optimizing nutrient availability for plant growth.

In this paper, we study the fractional pendulum oscillator equation. We solve it both numerically using finite differences method and analytically. The analytical solution is obtained by replacing the original oscillator using an integer-order oscillator. We also compare the numerical solution with the anlalytical solution obtained making use of the Chebyshev-homotopy method (KBM).

This study proposes a new method that utilizes the correlation structure between the number of words in the mail and the Bayesian score. Spam mails usually do not have a stable style and features. Spammers who send such mails, go on changing the features. The most often used statistical filter for email filtering is the Naive Bayesian filter. However, the training data and the word corpus that the filter designer utilized will determine the filter's design.  A new mail with unknown nature is classified into spam (unsolicited mail) or ham (legitimate mail) basing on a score by combining conditional probabilities of tokens in the mail. The statistical behavior of this score indicates some interesting features, which can be explored to improve performance of the filter. We report the results of an experiment using Enron data set and highlight the advantages of the new filter. We also propose a new method of testing the model using random data sets.

Graph coloring has many applications including task scheduling, parallel computation, frequency allocation, Network design, register allocation etc.  Here we concerned about the problem of vertex coloring on scheduling or assigning where these graph coloring provide algorithm to prevent or at least minimize the conflict scheduling. The objective is to provide simple way for solving worker scheduling problem in big industries. To overcome these problems we can schedule the task for workers by applying the concept of vertex coloring of graph theory using GRIN software.

This work presents a new solution for medical image classification using the Neural Network (NN) and Wavelet Network (WN) based on the Fast Wavelet Transform (FWT) and the Adaboost algorithm. The deep learning is a set of algorithms of machine learning, seeking to model with the abstractions of top level within the data using the architectures of models composed of multiple not linear transformations. This method is divided in two stages: The learning stage and the classification stage. The first consists to extract the features using the FWT based on the Multi Resolution Analysis (MRA). These features are used to calculate the inputs of the hidden layer. Then, those inputs are filtered by using the Adaboost algorithm to select the best ones corresponding to each image. The second consist to create an Auto Encoder (AE) using the best-selected wavelets of all images. Then, For the classification phase, our Convolutional Deep Neural Wavelet Network (CDNWN) architecture is obtained by applying a pooling for each hidden layer following a succession of Stacked AE. Our approach yielded classification rates that clearly outperform those stated in this publication. Our studies were conducted on two distinct datasets.

The study is devoted to the nature of blood flow with stenosis. The stenosis is modeled as a porous medium in order to apply the Rakhmatulin filtration model. Based on the penetrating model, considering the movement of particles immobile (immobile particles form stenosis), an equation describing the movement of blood with stenosis was obtained. Outside of stenosis, the equations turn into the Navier-Stokes equation. The resulting equations were solved numerically using the control method.The influence of the Reynolds number and the degree of stenosis on the nature of the flow on the behavior of blood is investigated.

This chapter deals with the impact of meteorological drought on groundwater in a heterogeneous aquifer system located in the plain southeast of Xanthi, northeastern Greece, using fuzzy set theory. The research aims to investigate the linear relationship between meteorological drought and groundwater levels of shallow wells, as well as the key role of geology in drought propagation, in terms of time, to groundwater.

The study area is bounded to the east by the Lake Vistonida, to the north by the River Kosynthos, to the west by the settlements of Vafeika and Genisea, and to the south by the hill of Avdira and the settlement of Nea Kessani. It is irrigated by a substantial number of wells (800 or more), predominantly shallow. The area exhibits strong heterogeneity in soil composition, consisting of coarse-grained fluvial deposits in the northwest and fine-grained lake sediments in the southeast.

To achieve the study's objective, fuzzy regression analysis based on Tanaka's approach is implemented. Tanaka's fuzzy linear regression model is reduced to a linear programming problem with inclusion constraints, ensuring that all observed data is included within the produced fuzzy band. The Standardized Reconnaissance Drought Index (RDI^St) and Standardized Precipitation Index (SPI) serve as inputs (independent variables) of the multiple fuzzy linear regression model. These drought indices are used to estimate drought intensity and are calculated for several reference periods. Water table serves the dependent variable, recorded by two water table loggers installed in different geological formation areas. The multiple fuzzy linear regression model produces a water table estimate for each time step with intervals of values assigned, each associated with a membership degree (fuzzy output). Hence, each fuzzy water table estimate is a fuzzy number characterized by its left boundary, right boundary, and central value, thus, effectively incorporating inherent uncertainty in estimation of groundwater level. To address potential inconsistencies, a modification of Tanaka's fuzzy linear regression model is suggested. It involves adding constraints to ensure that the central values of the fuzzy outputs are equal to or greater than their corresponding fuzzy semi-widths. In the evaluation process, the chapter employs three suitability measures and a comparison measure for fuzzy numbers. These measures assess the performance of the multiple fuzzy linear regression models and provide insights into the relationship between meteorological drought indices and groundwater levels.

The findings indicate a significant influence of annual meteorological drought on groundwater. Moreover, groundwater that are hosted in fine sediments, best related to annual drought indices, highlighting the crucial role of geology in drought propagation time. Furthermore, the use of RDI^St indices enhances the results of multiple fuzzy linear regression models in all cases, confirming the index's suitability for drought intensity estimation in semi-arid regions. The proposed modification to Tanaka's fuzzy linear regression model improves its applicability and reliability in the specific study area, providing insights for groundwater fluctuation.

In summary, the study contributes to understanding the complex interactions between meteorological drought, geology, and groundwater response in the investigated aquifer system.

The theory of graph is an extremely useful tool for solving combinatorial problems in different areas such as geometry, algebra, number theory, topology, operations research, and optimization and computer science. In this paper, the definitions of strong arc, non- strong arc, semi \(\mu\)- strong arc, semi \(\rho\)- strong arc, effective arc, semi \(\mu\)-effective arc and semi \(\rho\)-effective arc of intuitionistic fuzzy graph. Further, the types of domination on intuitionistic fuzzy graph by for strong arc and effective arc are defined. Also the non-strong arc in intuitionistic fuzzy (IF) Path and IF Cycle are respectively compared with Fuzzy Path and Fuzzy cycle. Finally the relation between the effective arc domination and strong arc domination of IF graph with a suitable illustration is given. The strong arc domination and effective arc domination are discussed and it is concluded that the strong arc domination is always less than or equal to the effective arc domination.

Aims: For various models of three-dimensional fluid which describe the flows in the Atmosphere and the Ocean in the gravity field with the stratification of vertical density, we investigate a relation between the essential spectrum of normal vibrations of internal waves and non-uniqueness of the limit amplitude of vibrations induced by external mass forces. To make the study more detailed and descriptive, we find the explicit spectrum for some particular domains.

Methodology: Fourier Transform, Spectral Analysis of Self-Adjoint Operators in Hilbert Spaces.

Results: We establish a direct relation between the frequency of the induced vibrations, the essential spectrum, and the non-uniqueness of the limit amplitude. We also find the explicit eigenfunctions and the eigenvalues of the spectrum for rectangular, spherical and cylindrical domains.