Recent Advances in Mathematical Research and Computer Science Vol. 7

An n X n matrix over a commutative ring with identity is clean if it is the sum of an idempotent matrix and a unit. This chapter discusses the required and sufficient criteria for a matrix A=\(\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in M_2 ({\displaystyle \mathbb {Z} })\) to be clean. 

The dynamics of quadratic polynomials have already gotten a lot of attention. The dynamics of rational functions and their properties are equally interesting. In this chapter, we consider the solution of the classical Yang-Mills equation by Witten [1] and generate computer images from a C++ computer program. We then use predictive modelling software to create an artificial neural network model based on RMS kind of error from two samples of points extracted from the generated images. By inputting the real parts of sample I and sample II [2] to the artificial neural network, the imaginary component of sample II could be predicted.  In forecasting the imaginary component of sample II, the real part of sample II is more important than the actual part of sample I. Sample II's projected imaginary component is then imported into Matlab Signal Processing Tool (SPTool) via Matlab workspace. We employ a stable band pass filter to the model to remove noise from it for its analysis. A modulated signal is produced, revealing that the methodology can be used to investigate the properties of the computer generated images derived from the generated wavelet. We then import the predicted imaginary part of sample II into autoSIGNAL software for time-frequency analysis of the continuous wavelet transform. This study attempts to bridge the gap between the dynamics of rational maps and the dynamics of electric the power system [3,4].

Fuzzy logic is a new and innovative technique that was used to develop an engineering control realization. In recent years, fuzzy logic has demonstrated its immense promise, particularly in the automation of industrial process control, where it allows the control design to be developed based on expert experience and experimental results. The projects that have been completed show that using fuzzy logic in technical process control has already resulted in better decisions than using standard control techniques. Fuzzy logic allows for the creation of an advisory system for decision-making that is based on operator experience and experiment results rather than a mathematical model. The current research focuses on a specific technological process: developing a support decision-making information system for the operational control of a lime kiln using fuzzy logic and a related expert-objective knowledge base.

Let G = (V, E) be a simple connected graph with p vertices and q edges. If G1, G2, …, Gn are connected edge disjoint subgraphs of G with E(G)=E(G1) \(\displaystyle \cup\) E(G2)\(\displaystyle \cup\) … \(\displaystyle \cup\)E(Gn), then (G1, G2, …, Gn) is said to be a decomposition of G. A decomposition (G1, G2, …, Gn) of G is said to be continuous monotonic decomposition (CMD) if each Gi is connected and |E(Gi)|=i, for every i=1, 2, 3, …, n. In this chapter, we introduced the concept Odd Star Decomposition of Lobster graph L and discussed several theorems based on diam (L).   

Generally the classification or segmentation refers the partitioning of an image into smaller regions to identify or locate the region of abnormality. Even though remains owing to the complex structure of brain tumors is challenging task in medical applications, due to contrary image, local observations of an image, noise image, and non uniform texture of the images and so on. Many techniques are available for image segmentation, but still it requires introducing efficient, fast medical image segmentation methods. This research article introduces an efficient image classification method based on K-means clustering integrated with a k- means clustering algorithms. The main contributions of this research paper are as follows: firstly combined Hybrid PSO-WCA (Particle Swarm Optimization-Water Cycle Algorithm) based Radial Basis Function Neural Network (RBFNN) machine learning classification model for brain tumours was used. Secondly, feature extraction, the GLCM (Gray Level Co-occurrence Matrix) technique was used. Thirdly, the malignant and benign tumours were classified using Fast fuzzy c-means, KNN (Nearest Neighbor) algorithm, Fuzzy c means algorithm, and K-Means algorithm with features as input for visual localization, and the performance of the clustering classification was presented. Finally, the proposed hybrid model PSO-WCA-RBFNN extracted tumor images are segmented to obtain 99.62 percent accurate representations of brain tumors. In addition, the proposed algorithm was compared with other current segmentation algorithm models PSO-RBFNN, WCA-RBFNN, and LMS-RBFNN are also presented. The results show that the proposed algorithm performs better in terms of accuracy. The experiment is conducted over 255 MRI brain images from Harvard Medical School.

Classification occurs when we are given a data set in which each data point is assigned a class based on the values and or characteristics of attributes. The k-Nearest Neighbor (kNN) algorithm in machine learning is a very simple and powerful tool for doing this. It is based on the idea that data points of a certain class are neighbours to one another. To find the class to which a given test data or unknown data belongs, one measures the Euclidean distances of the test data or unknown data from all the data points of all the classes in the training data in kNN.  Then, out of the k nearest distances, where k is any number greater or equal to 1, the class to which test data or unknown data is closest its most number of times is the class assigned to the test data or unidentified data. In this chapter, I propose a variation of kNN, which I call the ANN method (Alternative Nearest Neighbor) to distinguish it from kNN. The definition of neighbour is the distinguishing feature of ANN that distinguishes it from kNN. In ANN, the class to which the unknown data is neighbour is the class whose maximum Euclidean distance from its data points is less than or equal to the maximum Euclidean distance between all of the class's training data points.As a result, ANN will always provide a unique solution to each unknown data. In contrast, the solution in kNN may vary depending on the value of the number of nearest neighbours k. So, in kNN, as k is varied the performance may vary too. However, this is not the case with ANN; its performance for a specific training dataset is unique.

The main motivation behind finding the ANN machine learning method has been to modify the conventional kNN method in such a way that it is independent of the parameter k and the user of the method need not make a choice of k neighbors based on experience or other criteria.

For the training data [1] considered in this paper, the ANN gives 100% accurate result.

Modern data communication integrates traffic to carry audio, data and video simultaneously, enabling networks to accommodate an increasingly complex set of data traffic. Broadband Integrated Services Digital Network (B-ISDN) supports all types of traffic, including still and full-motion image applications [1]. With increasing number of wireless users, multi-media transmission makes network traffic more congested and there is need for effective traffic management to profit data transmission. Though, personal computing facilitates easy manipulation and secured storage, the access and exchange of information in wireless communication requires fast interconnections for profitable resource-sharing. With congestion as major challenge in broadband communication, transmission of multimedia data frames results in traffic levels overwhelming the network medium. Technically, this paper proffers an integrated scheme - link-to-link rate-based traffic flow control as basic technique to maximise bandwidth utilization, whereby all communications provided on traffic adopts the economy of sharing. Notably, implementation of Asynchronous Transfer Mode (ATM) technology and associated techniques delivers appreciable improvement in service quality. Also, improved continuity of service as users move maximally outside coverage areas were evidenced and modified transport scheme provided for negotiable Quality of Service (QoS). With the scheme mathematically implemented in simulation techniques, overlapping network traffics were managed efficiently and allotted bandwidth highly optimized. The traffic flow control proposed scheme implemented Additive Increase Multiplicative Decrease (AIMD) algorithm on transmitted packets to improve service quality. Therefore, capacity maximization of bandwidth scarce resource was achieved for all integrated services.

This study revives an approach to elicit human preferences based on the stimuli-response procedure long forgotten. The so-called school of Psycho-Physics (Weber-Fechner, 1860) [1], sought to make mathematical sense of the procedure above. Fifty years ago a new theory, Elementary Catastrophe Theory, (E.C.T.) [2], unfolding a unique Potential in our brain, provided the underlying dynamics needed to fulfill all the desiderata then missing. This axiomatization of a self-measurement process brings a rationale to the empirical data away from any “a priori” assumption about human purpose. Besides fitting the major landmark criteria in the fields of Value and Utility, like Arrow Impossibility result, this 5th degree symmetric polynomial exhibit a characteristic (Negative Schwarz’ Derivative) [3] going a long way to solve controversies and remove roadblocks in the progress of Portfolio Theory [4]. In the Annex we reproduce a long needed extension to Pratt’s [5]. Finally with this newly-found Human Scale (Cardinal) [6], grounded solely on the axiom of Non-Satiation, Free Energy could dislodge Entropy as a paradigm in this field [7].

It is shown that equations of mathematical physics can generate physical structures. From the equations of mathematical physics are obtained the closed inexact exterior form and associated closed dual form, which form a differential-geometrical structures. Such differential-geometrical structures describe the physical structures. The process of the emergence of physical structures, as shown, proceeds spontaneously under realization of any degrees of freedom.

This paper evaluates the performance of the bi-cubic B-spline collocation method. The bi-cubic B-spline collocation method is defined by using the recursive form of the bi-cubic B-spline basis function as basis functions. The suggested bi-cubic B-spline collocation method for Poisson's equations with Dirchlet's boundary condition problems is tested for viability and convergence. Absolute Relative Error analysis is also done by comparing with the exact solution for the convergence of the present method