International Journal of Mathematics And its Applications

2-12

Authors: V. R. Kulli

A graph index is a real number that is derived from molecular graphs of chemical compounds. In this paper, we introduce the multiplicative total reduced index, multiplicative reduced inverse degree, multiplicative reduced zeroth order index, general multiplicative reduced first Zagreb index of a graph and compute exact formulas for certain networks for chemical importance such as silicate networks, chain silicate networks, hexagonal networks, oxide networks and honeycomb networks.

17-25

Authors: M. Prasad, S. R. Nayaka and K. V. Prasad

Let $B_{2, 5}(n)$ denote the number of $(2, 5)$-regular bipartitions of a positive integer $n$ into distinct parts . In this paper, we establish several infinite families of congruences modulo powers of $2$ for $B_{2, 5}(n)$. For example, \begin{equation*} \sum_{n=0}^{\infty}B_{2, 5}\left(2^{2\alpha+1}\cdot5^{2\beta}n+\frac{2^{2\alpha+2}\cdot5^{2\beta}-1}{3}\right)q^n \equiv 2f_1f_{5}^3 \pmod{2^2}, \end{equation*} for $\alpha, \beta \geq 0$.

39-48

Authors: Dragan Obradovic, Lakshmi Narayan Mishra and Vishnu Narayan Mishra

The main factor in the development of human resources is knowledge and modern knowledge, through the application of which the individual and the company survive as competitive on the market. We acquire knowledge both during the formal educational process and through non-formal learning, in schools and colleges, and later in companies, in a traditional and online environment. Today, in science, great attention is paid to e-learning (e-learning) and e-education (e-education) using modern information and communication technologies. Nowadays, with the daily expansion of scientific knowledge there is also a huge progression of technology. In parallel, they developed technology in the field of education. Unfortunately, in our area this development and applications are flowing at a much slower pace (than needs dictate) in relation to the developed countries of the world and their educational systems. The main intention and goal of this paper is to point out on this occasion as well the importance, need and necessity of far broader and deeper study, and the application of modern educational technology at all levels of education, with special with an analytical review of contemporary university education.

55-62

Authors: Shradha Dwivedi, Sabir Ali Siddiqui, Peeyush Dwivedi and Sanjay Jain

The present paper deals in Bayesian analysis of the new distribution which is renamed here as SD-distribution. Bayesian estimates of the expected life, reliability function, hazard rate function is studied mathematically for the distribution, which is already established as a failure model. Similarly Bayesian estimates for the order statistics is also found here. The scale and shape parameters had been considered as random variables separately with certain prior distribution for the purpose of Bayesian estimation.

69-80

Authors: Ashok V. Bingi

In this paper, we introduce $\phi$-$\delta$-primary elements in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a counter example, it is shown that a $\phi$-$\delta$-primary element of $L$ need not be $\delta$-primary and found conditions under which a $\phi$-$\delta$-primary element of $L$ is $\delta$-primary.

91-95

Authors: Pradeep Kumar Dwivedi

In this paper we have to prove a common fixed point theorem for two pairs of compatible maps in dislocated metric space which generalizes and improves similar fixed point results.

101-104

Authors: Wanjala Victor and A. M. Nyongesa

In this paper, we introduce the class of (n,m)-metrically equivalent operators which is a generilazation of metrically equivalent operators and n-metrically equivalent operators. We then look at some properties of this class and its relation to some higher classes like quasi-isometries and the (n,m)-class (Q) operators. we also look at the relationship between this class and other equivalence relations like metrically equivalent and n-metrically equivalent operators.

111-113

Authors: Wanjala Victor and A. M. Nyongesa

In this paper, we introduce a new class of operator, the class of $(\alpha,\beta)$-Class (Q) operator acting on a complex Hilbert space $H$. An operator $T \in B(H)$ is said to be $(\alpha,\beta)$-Class (Q) if $\alpha^{2}T^{*2}T^{2}\leq (T^{*}T)^{2}\leq \beta^{2}T^{*2}T^{2}$ for $ 0 \leq \alpha\leq 1 \leq \beta$. We look at some properties that this class are priviledged to enjoy.

9-18

Authors: Vedanti Bhargava

Accurate and efficient model development in the field of evolutionary biology is becoming increasingly important to study phylogenetic relationships. Studying these phylogenetic relationships can provide greater insight into areas such as drug development, analyzing disease transmission, and studying the host-pathogen evolutionary relationship. This study uses the idea of a transition matrix to predict how long ago two species shared a common ancestor. Using this concept and various linear algebraic ideas, a mathematical model that simulated amino acid mutations was developed, and it was implemented in Python to perform trials. The main result of the development of the model was a matrix containing the expected values for the number of years ago two amino acids were the same. Simulations of the model can be run in Python multiple times and the average of the values obtained gives an estimate of how many years ago two amino acids were the same. These represent the number of years ago the species shared a common ancestor. The result varies based on the species data analyzed. This model provides a foundation for development of more complex models that enhance phylogenetic tree development. As stated previously, this can have substantial impacts on biotechnology and biomedical sciences, which often rely on analyzing these evolutionary relationships.

27-33

Authors: Mitesh J. Patel and G. V. Ghodasara

An injective function $f:V(G)\rightarrow{}\{1,2,\ldots,|V(G)|\}$ is said to be permutation labeling if each edge $uv$ is assigned with label $\Perm{f(u)}{f(v)}= \ \frac{(f(u))!}{| f(u)-f(v)|! } \ (f(u)>f(v))$ are all distinct. A graph which admits permutation labeling is called permutation graph. In this paper we prove that wheel graph, restricted square and degree splitting graph of bistar graph are permutation graphs. We also proved that arbitrary super subdivision of path graph, star graph and cycle graph are permutation graphs.

41-50

Authors: F. F. Ngwane

Test information function (TIF) plays an important role in designing a test. The TIF does not only depend on the logistic model such as the 1PL, 2PL and 3PL logistic models, but it also on the parameter distributions of the model. In this study we conducted simulations using the R statistical software on different examinee sizes and different exam length, and we compare the results when using uniform parameter distribution and normal parameter distribution. The findings of this study illustrate the following: First, when using item parameters that are all uniformly distributed the obtained test information function is significantly small compared to the test information function obtained when we use item parameters that all follow a normal distribution. Second, the test information function increases as the length of the exam increases when we use item parameters that are all normally distributed. Third, this study also suggest great flexibility in writing an exam as we can choose the item parameter distribution depending on the goals of the test.

63-65

Authors: V. Durai Murugan, R. Seethalakshmi and P. Namasivayam

In this paper, we verify the complementedness in the subgroup lattices of the group of $2 \times 2$ matrices over $Z_{11}$.

73-82

Authors: Annela R. Kelly

In combinatorial game theory, multi-player games expands on the research of two-person games and are widely applicable to computer science and business. The given paper studies the mis\`{e}re version of the subtraction game (Nim with one pile of counters) for two alliances extending the results of Kelly in \cite{2,3}; Liu and Zhao in \cite{5}, Liu and Wang in \cite{6} and Suetsugu in \cite{7}. The main result of the current paper determines the optimal distribution of the players among two alliances that minimizes $N$, the smallest number of counters such that the larger alliance has a winning strategy for all the games with more than $N$ counters.

95-104

Authors: V. Mohan Babu and C. Pandurangappa

An analysis is made to study a three-dimensional boundary-layer flow of non-newtonian fluid over a porous media. The governing partial differential equations of momentum are transformed into self-similar non-linear ordinary differential equations by using new form of suitable similarity transformations. The obtain equations are then should by using shooting method. The flow phenomena is characterized by the power-law index `n', permeability of porous media `$\omega $' porosity parameter `$\phi $' effective viscosity `$\lambda $', three-dimensionality parameters `$\alpha $'. The numerical results of the velocity profiles are obtained and displayed graphically.

113-117

Authors: Meera Meghpara, H. M. Karavadiya and V. J. Kaneria

In this paper, we have derived a new graph named spade graph. We have proved two copies of spade graph joined by an arbitrary path is $\alpha-$graceful, $n$ copies of spade graph joined by any vertex is $\alpha-$graceful, path union of $n$ copies of spade graph is $\alpha-$graceful and open star of finite copies of spade graph is also $\alpha-$graceful.

13-19

Authors: Ledi Wang

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.

29-33

Authors: M. Karthikeyan, L. Girija and P. K. Kishore Kumar

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.

41-47

Authors: H. Manjunatha and A. B. Madhava

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.

55-57

Authors: Brijesh Pratap Singh

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.

69-73

Authors: Puttaswamy, M. N. Prakasha and S. R. Nayaka

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.

85-90

Authors: Vidya S. Umadi and A. M. Sangogi

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.

95-101

Authors: Geoffrey Wu

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.

7-32

Authors: R. A. Mundewadi and B. A. Mundewadi

Biorthogonal spline wavelet method is proposed for the numerical solution of Linear and nonlinear integral and integro-differential equations. Biorthogonal spline wavelet filter coefficients have the prolongation and restriction operators. The performance of the proposed method is better than the existing ones in terms of super convergence with low computational time. Some of the test problems are demonstrated for the applicability and efficiency of the scheme.

41-47

Authors: S. G. Shete and R. G. Metkar

In this research paper the existence of minimal and maximal solution for Nonlinear Volterra type random integral equations is proved under some contraction, continuity and monotonicity conditions.

55-67

Authors: Kanchan Barman and Subhashish Biswas

In the paper, we introduce a new concept of soft contraction of Wardoski type which is a generalization of Banach contractive condition and prove a soft fixed point theorem which generalizes the Banach contraction principle in different ways. We also give some examples which show the validity of our results.

77-88

Authors: E. Azuaba, J. M. Orverem, Y. M. Kura and U. Jnr. Dahiru

An $S-L-B-I-Q-T$ epidemic mathematical model incorporating drug therapy and treatment is investigated for malaria disease. We obtained the Disease Free Equilibrium (DFE) points and compute the basic reproduction number ($R_{0} $). The local and global stability of the Disease Free Equilibrium was analyzed using Jacobian matrix stability techniques and Lyapunov function respectively. The local and global stability was asymptotically stable if $R_{0} <1$ and $R_{0} \le 1$ respectively. Sensitivity analysis of $R_{0} $ for drug therapy and treatment showed that $R_{0} $ is strictly a decreasing function of $\sigma _{3} ,\theta ,\nu ,\tau $ and $p$. The numerical simulation of $R_{0} $ and control parameters of the model were presented graphically. The findings of this study strongly suggest a combination of effective drug therapy and treatment as a crucial strategy to control the malaria disease.

95-99

Authors: Ram Milan Singh

In this paper, we generalized the method to calculating the square root of matrix whose characteristic is quadratic and how to Cayley-Hamilton theorem may be used to determine the formula for all square root of matrix whose order is $2\times2$.

9-14

Authors: Khawlah Hashim Hussain

In this paper, the nonlinear Volterra integral equations are solved by using new modifications of the Adomian decomposition technique. The approximate solution of this equation is calculated in the form of a series with easily computable components. The accuracy of the proposed techniques are examined by comparison with exact results. The test problems are presented to illustrate the reliability and performance of the techniques.

23-30

Authors: S. M. Vanithalakshmi, S. K. Narasimhamurthy and M. K. Roopa

In this paper, we determined the fundemental tensor fields $( \tilde g_{ij}, \tilde g_{i\bar j})$ and inverse of these tensor fields, their determinant. Further, we studied some properties of non-Hermitian $\mathbb{R}$-complex Finsler space with Matsumoto metric.

41-52

Authors: Jula Kabeto Bunkure

The functional-analytic approach to the solution of partial differential equations requires knowledge of the properties of spaces of functions of one or several real variables. A large class of infinite dimensional dynamical systems (evolution systems) can be modeled as an abstract differential equation defined on a suitable Banach space or on a suitable manifold therein. The advantage of such an abstract formulation lies not only on its generality but also in the insight that can be gained about the many common unifying properties that tie together apparently diverse problems. It is clear that such a study relies on the knowledge of various spaces of vector valued functions (i.e., of Banach space valued functions). For this reason some facts about vector valued functions is introduced. We introduce the various notions of measurability for such functions and then based on them we define the different integrals corresponding to them. A function $f:\Omega \to X$ is strongly measurable if and only if it is the uniform limit almost everywhere of a sequence of countable-valued, $\Sigma$-measurable functions. If X is separable and $f: \Omega\to X$, be a function then the following three properties are equivalent: (a) f is strongly measurable; (b) f is weakly measurable; (c) f is Borel measurable. A strongly measurable function $f: \Omega\to X$, is Bochner integrable if and only if the function $\omega\to \left\|f\left(\omega \right)\right\|_X$ is Lebesgue integrable (i.e., $\left\|f\left(.\right)\right\|_X\in L^{1}(\Omega)$). The emphasis of the project is on the so-called Bochner integral, which generalizes in a very natural way the classical Lebesgue integral to vector valued functions. We continue with vector valued functions and introduce the so-called Lebesgue-Bochner spaces, which extend to vector valued functions of the well known Lebesgue $L^{p}$-spaces. We also consider evolution triples and the function spaces associated with them. Evolution triples provide a suitable analytical framework for the study of a large class of linear and nonlinear evolution equations.

59-68

Authors: Sanaa Al-Marzouki

A new generalization of the power Lomax (PL) distribution, called exponentiated power Lomax (EPL), is introduced for modeling lifetime data. Some statistical properties of the EPL distribution are provided. Explicit expressions for the quntile, moments, moment generating function, probability weighted moments and order statistics are studied. Maximum likelihood estimation technique is employed to estimate the model parameters are presented. In addition, the superiority of the EPL distribution is illustrated with applications to two real data sets.

85-92

Authors: Sanaa Al-Marzouki and Taghreed Al-Said

A new three-parameter lifetime model, called the Truncated Weibull Lomax (TWL) distribution, is introduced. We investigate the density (pdf), distribution function (cdf), survival function (sf), hazard function (hrf), quantile function, r\textsuperscript{th} moment, incomplete moments and order statistics. Maximum Likelihood methods to estimate the TWL distribution parameters are studied. One real data set is applied to illustrate the flexibility of the TWL model compared with some Known models.