Current Topics on Mathematics and Computer Science Vol. 3

A harmonious labeling on a graph G is an injection from the vertices of G to the group of integers modulo k, where k is the number of edges of G, that induces a bijection between the edges of G and the numbers modulo k by taking the edge label for an edge ab to be the sum of the labels of the two vertices a\(\cong\)b(modk).In this paper we prove that the symmetrical trees of diameter four admit harmonious labeling. Harmonious graphs can be used in the study of error correcting codes and channel assignment problems.

In \(\mathbb{R}\), the Brouwer’s fixed point theorem states that for any continuous functions \(\mathit{f}\) : [0,1] \(\rightarrow\) [0,1] has a fixed point. There is observing the nature of its functions, the domain of the function, or a support function. In this article, we show that the derivative function on [0,1] into itself has a fixed point even though the derivative function does not necessarily continuous.

Drastic growth of sensor network technology paves way to the socio economic applications in all aspects. Similarly introduction of new sensor for various parameters increases the scope for data collection and manipulation for useful inferences. In order to exploit largest unutilized resources vested with the ocean environment, the necessity arises for the researchers to find out the sensor network suitable for the monitoring of fish movements and to decide on the probable area for data communication in underwater. Main objective of the paper is to monitor and analyze the fish movement and behaviour in the underwater sensor networks by introducing the new algorithm called AFISH [ARTIFICIAL FISH] algorithm, which will study the movement and behaviour of fish in the water with certain parameters. The behaviour is observed with certain under water sensors placed along the Length, breadth and depth of earmarked area and with the help of data fusion the information is observed and manipulated to understand the data communication area and optimum time for the same [1]. The simulation results show that the proposed AFISH algorithm for fish movement monitoring works effectively under certain presumed conditions.

Title: An advanced approach with focused crawling for various anchor texts is discussed in this paper.

Background: Most of the search engines search the web with the anchor text to retrieve the relevant pages and answer the queries given by the users. The crawler usually searches the web pages and filters the unnecessary pages which can be done through focused crawling. A focused crawler generates its boundary to crawl the relevant pages based on the link and ignores the irrelevant pages on the web.

Methods and Findings: In this paper, an effective focused crawling method is implemented to improve the quality of the search. Here, three learning phases are considered namely, content-based, link-based and sibling-based learning are undergone to improve the navigation of the search. In this approach, the crawler crawls through the relevant pages efficiently and more relevant pages are retrieved in an effective way.

Study Objective: The objective of the study is that more number of relevant pages is retrieved for different anchor texts with three learning phases using focused crawling.

The system of coupled nonlinear differential equations that governs yarn motion in general is derived. Because the equations are expressed in a (non-uniformly) rotating observation frame, they can be used to describe yarn unwinding from stationary packages.  We discuss the physical implications of virtual forces that occur in non-inertial frames, with a focus on a lesser-known force that only arises in non-uniformly rotating frames. When the unwinding point is near the package's edges, the quasi-stationary approximation is invalid since the angular velocity is changing with time, we show that this force should be considered. In this transient regime, when yarn movement becomes complex, the additional force has an effect on yarn dynamics, which can lead to yarn slippage and even breakage.

In many textile operations, yarn unwinding from a package is important. The efficiency of the procedure as well as the quality of the end product are directly influenced by the unwinding process's stability. The tension oscillates during the unwinding process.

This is most visible during over-end unwinding from a static package, as the yarn is removed at a high velocity in the package axis' direction. Even at very high unwinding velocities, the package's ideal shape allows for an optimal shape of the yarn balloon and low and consistent tension. The goal of this project is to write down the equations that explain yarn motion during unwinding and to build a mathematical model that can be used to replicate the process.

In this chapter, we deal with a continuous review perishable inventory system with a service facility having two heterogeneous servers (henceforth, referred to as server 1  and 2 server ) and server 2  is always available but server 1  goes on vacation when the inventory level is less than or equal to one. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The life time of each commodity is assumed to be exponential. The inventory is replenished according to an \(s, Q\) control policy and the replenishing times are assumed to be exponentially distributed. At the end of a vacation period, service begins if inventory level is more than one in the system. Otherwise, the server takes another vacation immediately and continuous in the same method until he finds at least two items in the system upon returning from a vacation. Any arriving primary customers who find the waiting room full are queued in the orbit. The inter-retrial times are exponentially distributed. The joint probability distribution of the number of customers in the waiting room, the number of customers in the orbit, the status of the servers and the inventory level are obtained for the steady state case. Some important system performance measures in the steady state are derived, and the long-run total expected cost rate is also calculated. The results are illustrated numerically.

Object-oriented software design is difficult, and reusable object-oriented software design is even more complicated. The design must be specific to the problem, but also general enough to be able to solve future requirements. In the process of developing an interactive computer application, design represents probably the most important factor that can affect its quality. Propitiously, there are presently various design patterns that can be used to solve a general problem.Competent architects' design solutions are represented through design models.As a result, in the oriented object programming paradigm, any design model seeks to solve a specific design issue. The purpose of this work is to propose an interactive computer application that can be utilised in the study of triangle geometry. The design of the computer application is improved by using of the "Factory" creational pattern and “Observer” behavioural pattern. The use of factory pattern binds parallel class hierarchies such that it can localize the knowledge of which classes belong together. The most important advantage of using observer pattern is that this template simplifies programming work considerably. The implementation is accomplished through the Java programming language which is an independent platform. The use of these two design patterns improves development, extensibility and reusability of triangle geometry computer application.

Researchers can employ parametric methods (if they exist) or non-parametric approaches (if acceptable parametric methods do not exist) to test the validity of assertions during their research. The Chi-Square (x²) distribution plays a significant role in both parametric and non-parametric methods and many of its most important applications are explored in this paper, to guide the researchers in its use. 

In this work, analogous to Molodtsov Soft sets, We define a new class of Soft sets, namely Soft A_RS closed sets and its complement Soft A_RS open sets. Further we compare the properties of Soft A_RS closed sets with Soft closed set, Soft semi closed set, Soft \(\alpha\) closed set, Soft JP closed set, Soft gsp closed set and we prove the concept of Soft A_RS Closed sets is independent of the concept of Soft \(\omega\) closed sets, Soft \(\alpha\)g closed set, Soft gs closed set, Soft gp closed set, Soft pre closed set, Soft Strongly g closed set, Soft g closed, Soft \(\beta\) closed set.
We show that the concept of Soft A_RS closed sets is stronger than Soft semi closed set and weaker than the concept of Soft generalized semi pre closed sets. The same is applicable for Soft A_RS open sets as well. We find the characterization of Soft A_RS closed sets and Soft A_RS open sets. We investigate fundamental properties of Soft A_RS closed sets discuss their properties.

A Latin square model can simultaneously control two sources of nuisance variability.  As an experimental design model, the Graeco-Latin square is an extension of a Latin square and can simultaneously control three sources of nuisance variability. The first mentioning of the Graeco-Latin square was in 1782. The following aspects of this model will be discussed:  A brief history, estimation and ANOVA, use for the analysis of experimental data (example with R code given), model generation and a test for non-additivity. An R example of the Hyper Graeco-Latin square model, which extends the Graeco-Latin square to controlling four sources of nuisance variability, will also be discussed.  

The negative binomial distribution can be seen as the waiting time distribution associated with the binomial distribution. In a similar fashion the negative hypergeometric distribution can be seen as the waiting time distribution of the hypergeometric distribution. A definition and summary of the main theoretical results of this distribution will be given. Results and outputs (using R) of applications of the negative hypergeometric distribution will be shown and discussed.

Among all the manifestations of chaos in various scientific fields, deriving Random Number Generators by the use of chaotic systems is widely investigated. Although the fundamental premise of the chaos theory states determinism in the resulting series, with the extreme sensitivity and dependence on the initial conditions of the system, one could address the problem of randomness in the realms of chaos theory. In this study, we present 8 defined mathematical schemes applying to the experimental data extracted from capacitors’ voltages of the classical configuration of Chua’s circuit. Each of the 8 suggested schemes operate as functions aiming for the generation of randomly distributed values and are eventually compared with the commercially common timer-based random generators. The Monte Carlo Integration Algorithm, the evaluator method of the research, indicates the spectral distributed data and then the ranking of the schemes has been proceeded through a visualization of the supposed algorithm. As the geometrical domain in the Monte Carlo Integration has defined in such a way that the most randomly scattered data set would result in a closer estimation of the number Pi, the suggested scheme, Frequency indicator, is evaluated as the highest-ranked scheme in that regard, with the estimated numerical value of 3.1424 for Pi.

In this paper different classification methods are compared using base and meta (Combination of Multiple Classifier for training) level classifiers, for the fruitful diagnosis of Lower Back Pain. Radiology based different imaging techniques are also compared for diagnosing Lower Back Pain, like Computed Tomography (CT)  scan, Magnetic Resonance Imaging (MRI).  The Lower Back Pain becomes chronic with age, so needs to be correctly diagnose with symptoms in the early age. Five independent classifiers were implemented at base level and meta level. At meta level, five combinations of different classifiers were implemented, using voting technique. According to the scores, the overall classification using Naïve Bayes and Multilayer Perceptron got the maximum efficiency 83.87%. The purpose of this paper is to diagnose healthy individuals efficiently. To carry out study the Lower Back Pain Symptoms Dataset is used from very famous platform for predictive modelling, Kaggle. The experiments were carried out in WEKA (Waikato Environment for Knowledge Analysis), suite of machine learning software [1].