Earthline Journal of Mathematical Sciences <p style="text-align: justify;">The Earthline Journal of Mathematical Sciences (e-ISSN: 2581-8147) is a peer-reviewed international journal devoted to publishing original research papers containing substantial contributions in any direction of Pure and Applied Mathematics, Theoretical and Applied Statistics, Theoretical Computer Science, Mathematical Physics, Mathematical Chemistry, Mathematical Modelling, Mathematical Biology &amp; Biostatistics. Applications of Mathematics in Finance and Economics are also welcome. .</p> Earthline Publishers, Madanambedu, Chittoor, Andhra Pradesh, India en-US Earthline Journal of Mathematical Sciences 2581-8147 <p><img src=""><br>This work is licensed under a&nbsp;<a href="" rel="license">Creative Commons Attribution 4.0 International License</a>.</p> On Generating Tridiagonal Matrices of Generalized (s,t)-Pell, (s,t)-Pell Lucas and (s,t)-Modified Pell Sequences <p>In this study, we define some tridigional matrices depending on two real parameters. By using the determinant of these matrices, the elements of (s,t)-Pell, (s,t)-Pell Lucas and (s,t)-modified Pell sequences with even or odd indices are generated. Then we construct the inverse matrices of these tridigional matrices. We also investigate eigenvalues of these matrices.</p> S. Uygun Copyright (c) 2021-08-12 2021-08-12 7 2 229 249 10.34198/ejms.7221.229249 Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials <p>Using the Al-Oboudi type operator, we present and investigate two special families of bi-univalent functions in $\mathfrak{D}$, an open unit disc, based on $\phi(s)=\frac{2}{1+e^{-s} },\,s\geq0$, a modified sigmoid activation function (MSAF) and Horadam polynomials. We estimate the initial coefficients bounds for functions of the type $g_{\phi}(z)=z+\sum\limits_{j=2}^{\infty}\phi(s)d_jz^j$ in these families. Continuing the study on the initial cosfficients of these families, we obtain the functional of Fekete-Szeg\"o for each of the two families. Furthermore, we present few interesting observations of the results investigated.</p> S. R. Swamy Copyright (c) 2021-08-18 2021-08-18 7 2 251 270 10.34198/ejms.7221.251270 The New Results in n-injective Modules and n-projective Modules <p>In this paper, we introduce and clarify a new presentation between the <em>n</em>-exact sequence and the <em>n</em>-injective module and <em>n</em>-projective module. Also, we obtain some new results about them.</p> Samira Hashemi Feysal Hassani Rasul Rasuli Copyright (c) 2021-08-19 2021-08-19 7 2 271 285 10.34198/ejms.7221.271285 Trifunction Bihemivariational Inequalities <p>In this paper, we consider a new class of hemivariational inequalities, which is called the trifunction bihemivariational inequality. We suggest and analyze some iterative methods for solving the trifunction bihemivariational inequality using the auxiliary principle technique. The convergence analysis of these iterative methods is also considered under some mild conditions. Several special cases are also considered.&nbsp; Results proved in this paper can be viewed as a refinement and improvement of the known results.</p> Muhammad Aslam Noor Khalida Inayat Noor Copyright (c) 2021-08-30 2021-08-30 7 2 287 313 10.34198/ejms.7221.287313 Detecting Electronic Banking Fraud on Highly Imbalanced Data using Hidden Markov Models <p>Recent researches have revealed the capability of Machine Learning (ML) techniques to effectively detect fraud in electronic banking transactions since they have the potential to detect new and unknown intrusions. A major challenge in the application of ML to fraud detection is the presence of highly imbalanced data sets. In many available datasets, majority of transactions are genuine with an extremely small percentage of fraudulent ones. Designing an accurate and efficient fraud detection system that is low on false positives but detects fraudulent activity effectively is a significant challenge for researchers. In this paper, a framework based on Hidden Markov Models (HMM), modified Density Based Spatial Clustering of Applications with Noise (DBSCAN) and Synthetic Minority Oversampling Technique Techniques (SMOTE) is proposed to effectively detect fraud in a highly imbalanced electronic banking dataset. The various transaction types, transaction amounts and the frequency of transactions are taken into consideration by the proposed model to enable effective detection. With different number of hidden states for the proposed HMMs, simulations are performed for four (4) different approaches and their performances compared using precision, recall rate and F1-Score as the evaluation metrics. The study revealed that, our proposed approach is able to detect fraudulent transactions more effectively with reasonably low number of false positives.</p> Abukari Abdul Aziz Danaa Mohammed Ibrahim Daabo Alhassan Abdul-Barik Copyright (c) 2021-09-06 2021-09-06 7 2 315 332 10.34198/ejms.7221.315332 Generalized Oresme Numbers <p>In this paper, we introduce the generalized Oresme sequence and we deal with, in detail, three special cases which we call them modified Oresme, Oresme-Lucas and Oresme sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.</p> Yüksel Soykan Copyright (c) 2021-09-13 2021-09-13 7 2 333 367 10.34198/ejms.7221.333367 f-Biharmonic Curves in the Three-dimensional Para-Sasakian Space Forms <p>In this paper, we give some characterizations for proper <em>f</em>-biharmonic curves in the para-Bianchi-Cartan-Vranceanu space forms with 3-dimensional para-Sasakian structures.</p> Murat Altunbaş Copyright (c) 2021-09-13 2021-09-13 7 2 369 379 10.34198/ejms.7221.369379 The Topp-Leone Weibull Distribution: Its Properties and Application <p>This paper presents a new generalization of the Topp-Leone distribution called the Topp-Leone Weibull Distribution (TLWD). Some of the mathematical properties of the proposed distribution are derived, and the maximum likelihood estimation method is adopted in estimating the parameters of the proposed distribution. An application of the proposed distribution alongside with some well-known distributions belonging to the Topp-Leone generated family of distributions, to a real lifetime data set reveals that the proposed distribution exhibits more flexibility in modeling lifetime data based on some comparison criteria such as maximized log-likelihood, Akaike Information Criterion [AIC=2k-2 log⁡(L) ],&nbsp;Kolmogorov-Smirnov test statistic (K-S)&nbsp;and Anderson Darling test statistic (A<sup>*</sup>) and Crammer-Von Mises test statistic (W<sup>*</sup>).</p> Diamond O. Tuoyo Festus C. Opone N. Ekhosuehi Copyright (c) 2021-09-17 2021-09-17 7 2 381 401 10.34198/ejms.7221.381401 New Families of Bi-Univalent Functions Governed by Gegenbauer Polynomials <p>The aim of this article is to initiating an exploration of the properties of bi-univalent functions related to Gegenbauer polynomials. To do so, we introduce a new families \mathbb{T}_\Sigma (\gamma, \phi, \mu, \eta, \theta, \gimel, t, \delta) and \mathbb{S}_\Sigma (\sigma, \eta, \theta, \gimel, t, \delta ) of holomorphic and bi-univalent functions. We derive estimates on the initial coefficients and solve the Fekete-Szeg &nbsp;problem of functions in these families.</p> Abbas Kareem Wanas Copyright (c) 2021-09-27 2021-09-27 7 2 403 427 10.34198/ejms.7221.403427 Burhan Distribution with Structural Properties and Applications in Distinct Areas of Science <p>In this work a novel distribution has been explored referred as Burhan distribution. This distribution is obtained through convex combination of exponential and gamma distribution to analyse complex real-life data. The distinct structural properties of the formulated distribution have been derived and discussed. The behaviour of probability density function (pdf) and cumulative distribution function (cdf) are illustrated through different graphs. The estimation of the established distribution parameters are performed by maximum likelihood estimation method. Eventually the versatility of the established distribution is examined through two real life data sets.</p> Aijaz Ahmad Muzamil Jallal S. Quratul Ain Rajnee Tripathi Copyright (c) 2021-10-06 2021-10-06 7 2 429 445 10.34198/ejms.7221.429445