Issue 4 Volume 8 (2020)

Authors : D. Pradeep
Title : Finite-Dimensional Approximation of Modified Laverentiv Method for Nonlinear Ill-posed Operators
Volume, Issue, Year : 8(4)(2020)
Pages : 1−11

Abstract: In \cite{Pr1}, the authors have suggested a modified iterative Laverentiv method for solving nonlinear ill-posed equations involving monotone operators. In this paper, we propose the finite-dimensional approximation of the scheme considered in \cite{Pr1}. We have shown the regularization parameter and the error estimates are optimal order. Finally, we also present some numerical results to verify the theoretical estimates.

Keywords: Nonlinear Ill-posed Problems, Regularization, Iterative Method, Parameter Choice Rule.

Authors : Ledi Wang
Title : Population of Fish: Mathematical Modeling of the Optimal Fishing Rate
Volume, Issue, Year : 8(4)(2020)
Pages : 13−19

Abstract: We consider a discrete mathematical model of the population size of fish in an isolated environment. This model includes the effect of fishing at a constant rate. We analyze the model and obtain the optimal fishing rate. This rate is optimal in the sense that the population maintains a certain size at this rate, but any larger fishing rate leads to the extinction of the fish population.

Keywords: Population of Fish, Mathematical Modeling, Optimal Fishing Rate.

Authors : Kalan Nguyen
Title : Cryptography and Number Theory
Volume, Issue, Year : 8(4)(2020)
Pages : 21−27

Abstract: Number theory is the branch of pure mathematics which studies positive whole numbers also called natural numbers and integers. The goal of number theory is to discover relationships between numbers that may not be apparent at face value. Cryptography is the study of encryption and decryption of information. By encrypting information the data can be transmitted to the intended recipient securely across the internet (insecure network). The use of numbers in cryptography is inextricably linked.

Keywords: Cryptography, Number theory, Encryption and decryption.

Authors : M. Karthikeyan, L. Girija and P. K. Kishore Kumar
Title : Set Domination Number of Jump Graph
Volume, Issue, Year : 8(4)(2020)
Pages : 29−33

Abstract: Let $G = (V,E)$ be a connected graph. A Set $D\subseteq V(J(G))$ is a set dominating set (sd-set) of jump graph if for every set $T \subset V(J(G))-D$, there exists a non empty set $S \subset D$ such that the subgraph $\langle S \cup T\rangle$ induced by $S \cup T$ is connected. The set domination number $\gamma_s(J(G))$ of $J(G)$ is the minimum cardinality of a sd-set of $J(G)$ .We also study the graph theoretic properties of $\gamma_s(J(G))$ and its exact values of some standard graphs. The relation between $\gamma_s(J(G))$ with other parameters is also investigated.

Keywords: Domination number, Set domination number, Jump graph.

Authors : T. J. Rajesh Kumar and Antony Sanoj Jerome
Title : Gaussian Prime Labeling of Super Subdivision of Star Graphs
Volume, Issue, Year : 8(4)(2020)
Pages : 35−39

Abstract: Gaussian integers are the complex numbers of the form a + bi where $a, b \in Z$ and $i^{2} =-1$ and it is denoted by $ Z[i]$. A Gaussian prime labeling on G is a bijection from the vertices of G to $[\psi_{n}]$, the set of the first n Gaussian integers in the spiral ordering such that if $uv \in E(G)$, then $\psi(u)$ and $\psi(v)$ are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of super subdivision of star graphs.

Keywords: Gaussian Integers, Gaussian Prime Labeling, Super Subdivision of Graphs.

Authors : H. Manjunatha and A. B. Madhava
Title : A Note on Discrete Fourier Transform
Volume, Issue, Year : 8(4)(2020)
Pages : 41−47

Abstract: This paper contains basic concept of DFT, The Inverse Discrete Fourier Transform, Computation of DFT directly via the FFT, DFT (or FFT) for spectral analysis, FFT of the sum of two sine-waves, Verification of Zero-padding, The even symmetry by Extension.

Keywords: Fourier Transform, Zero-padding, Spectral Analysis.

Authors : S. Mbala, M. M. Manene and J. A. M Ottieno
Title : Validation of Symmetric Tuth Detection Models Using Examination Dishonesty Research
Volume, Issue, Year : 8(4)(2020)
Pages : 49−53

Abstract: Examinations across the world play a vital role in the education system. Unfortunately, the process of examinations is greatly affected by the act of cheating hence giving a wrong impression on the results. The research on examination dishonesty among secondary school students is considered to be a sensitive issue, thus direct questioning techniques are not useful as the respondents will either refuse to answer the survey questions or even if they do, may give false answers for fear of being known to have been involved in the cheating. In this paper we have tested the validity of Symmetric truth detection models using examination dishonesty research. This research was done in a national Secondary school in Kenya. The results were used to validate the Symmetric Truth detection models by comparing the results with the results of the Asymmetric questioning technique. We have shown that the Symmetric truth detection models are more reliable when investigating sensitive information compared to Asymmetric truth detection models.

Keywords: Randomized response, Symmetric questioning technique, Sensitive questions, Asymmetric questioning technique, examination dishonesty.

Authors : Brijesh Pratap Singh
Title : Certain Transformation Formulae for Basic Hypergeometric Series
Volume, Issue, Year : 8(4)(2020)
Pages : 55−57

Abstract: In present paper we have taken certain known summation formulae due to Verma and Jain and by making use of Bailey's transformation an attempt has made to establish certain beautiful and interesting transformation formulae for q-Hypergeometric series.

Keywords: Basic Hypergeometric series, transformation, summation formulae.

Authors : P. R. Hampiholi, A. S. Joshi and J. P. Kitturkar
Title : On Spectra of D-Eccentricity Matrix of Some Graphs
Volume, Issue, Year : 8(4)(2020)
Pages : 59−68

Abstract: For any two vertices $u$ and $v$ of a graph $G$, $d(u, v)$ is the length of the shortest path between the vertices $u$ and $v$. D. Reddy Babu and P.L.N. Varma introduced the concept of D-distance. D-distance considers the degree of all vertices present in a path while defining its length. In this paper, D-eccentricity spectra of D-eccentricity matrix of some class of graphs are computed.

Keywords: Distance, D-distance, Eccentricity matrix, Spectrum.

Authors : Puttaswamy, M. N. Prakasha and S. R. Nayaka
Title : Isolated Domination Polynomial of a Graph
Volume, Issue, Year : 8(4)(2020)
Pages : 69−73

Abstract: A dominating set $ S$ of a graph $ G$ is called an isolated dominating set, if $\langle S\rangle $ contains at least one isolated vertex. The minimum cardinality of an isolated dominating set is called the isolated domination number, symbolized by $\gamma_o(G)$. In this paper, we initiate the study of a graph polynomial called isolated domination polynomial. We determine the isolated domination polynomial of some standard graphs and characterize some class of graphs through the isolated domination polynomial. Finally, we determine the isolated domination polynomial at some particular values.

Keywords: Dominating set, Isolated dominating set, Domination polynomial, isolated domination polynomial of a graph.

Authors : Shweta Wasnik and Subhashish Biswas
Title : Some Applications of Chatterjea - Pata Type Fixed Point Theorem in Modular Spaces
Volume, Issue, Year : 8(4)(2020)
Pages : 75−83

Abstract: In this paper, we prove Chatterjea - Pata type fixed point theorem in modular spaces which generalizes and improves some old results. Also we give an application for the existence of solutions of integral equations in modular function spaces.

Keywords: Fixed point, Modular spaces, Nonlinear integral equations.

Authors : Vidya S. Umadi and A. M. Sangogi
Title : Review Paper on Random Number Generators Used in Cryptosystems
Volume, Issue, Year : 8(4)(2020)
Pages : 85−90

Abstract: A variety of random number generator techniques has been developed during the past years and has been used in different cryptosystems. This paper reviews the existing methods of random number or sequence generator techniques and proposes a new technique using chaos theory.

Keywords: Random Numbers, Cryptosystems, Sequence generator techniques.

Authors : S. R. Nayaka
Title : Pendant Edge Domination in Graphs
Volume, Issue, Year : 8(4)(2020)
Pages : 91−94

Abstract: A dominating set of a graph $G$ is said to be pendant dominating set if the induced subgraph contains at least one pendant vertex. A subset $F$ of edges of $G$ is said to be an edge dominating set if every edge not in $F$ is incident to at least one edge in $F$. An edge dominating set is said to be pendant edge dominating set if an edge induced subgraph $\langle F\rangle$ contains an edge of degree one. In this article, we initiate the study of pendant edge domination of graph and compute exact values for some standard graphs.

Keywords: Pendant Domination, Edge Domination, Pendant edge domination.

Authors : Geoffrey Wu
Title : Modeling Bridge Oscillations Induced by Marching Soldiers
Volume, Issue, Year : 8(4)(2020)
Pages : 95−101

Abstract: This work simulates the effects of soldiers marching on a bridge, by first using Newton’s second law to write a differential equation describing the bridge's motion, and then solving the equation to yield a simple harmonic oscillator. Each of the soldiers’ steps is modeled as a constant impulse on the bridge. Computer simulations model the effects of between 5 and 200 soldiers marching both in and out of unison. We find that in both cases, the maximum amplitude of the oscillations of the bridge is proportional to the time the soldiers spent marching on the bridge. In the case where the soldiers march in unison, there is a linear relationship between the maximum amplitude of the oscillations and the number of marching soldiers. However, when the soldiers do not march in unison, the oscillations are proportional to the square root of the number of soldiers. Thus, it is preferable for soldiers to march out of unison on a bridge, since marching in unison could lead to a rapid increase in the amplitude of oscillations and the collapse of the bridge.

Keywords: Resonance, oscillation, bridge.

Authors : S. Singh and R. K. Srivastava
Title : Markovian Queueing System with Bulk Arrival and Three Types of Essential and Optional Services
Volume, Issue, Year : 8(4)(2020)
Pages : 103−112

Abstract: This paper analyses the single server bulk arrival Markovian queueing system where customers arrive according to the general bulk rule and follows the Poisson distribution. Customers are served with three types of services, one is essential service and two of them are optional services. Server is giving service to the customers according to Exponential distribution with the bulk service rule. After providing service to the bulk, server goes for a vacation according to the exponentially distribution. Server returns from the vacation when there are sufficient number of customers is available for receiving the service. Again server go back for a vacation if arrived number of customers is not enough to get the essential service. The server returns when arrived bulk is sufficient. After receiving the essential service, customers have the choice that they take first essential service, second essential service or take both. We have also include the impatience behaviour of customers i.e. the bulk of customers also balks from the system if the server is busy to giving service to other customers or server is on the vacation. We have computed the time dependent probability generating functions with the help of corresponding steady state results. Also we determine the number of customers waiting in the queue for this system.

Keywords: Bulk Arrival, Essential Service, Optional Service, Markovian Queueing System.