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.