Research Updates in Mathematics and Computer Science Vol. 4

In this article, we report for the first time the analytical solution for a rotational pendulum system. Analytical expression for the period of oscillations is provided. We also derive approximate analytical solution for the damped case.

We introduce new techniques for solving truly nonlinear oscillators. We give approximate expressions for their period. Using this period we find approximate analytical solutions using both trigonometric and jacobian functions. The results are illustrated with concrete examples.

In the present paper we give an elementary method in order to solve a delayed Duffing equation. The results are compared with the Runge-Kutta numerical solution.

In this paper we introduce a forced Kawahara equation. We solve it making use of a direct ansatz method.

We also solve it via splines method. Numerical examples illlustrate the accuracy of the obtained results.

Graph mining has become a well-established discipline within the domain of data mining. It has received much interest over the last decade as advances in computer hardware have provided the processing power to enable large-scale graph data mining to be conducted. Frequent subgraph mining (FSM) plays a very significant role in graph mining, attracting a great deal of attention in different domains, such as Bioinformatics, web data mining and social networks. Research on FSM started around 1994, but it has become popular since 2008 when the size of graphs in different domains became relatively large. Several techniques have been proposed in the literature for the FSM problem. In this paper, we reviewed some recently presented FSM techniques and investigated some challenges and future research directions. A few surveys have been conducted to review different techniques for the FSM problem. However, existing surveys highlighted only the methodology adopted for frequent subgraph discover but did not critically review their shortcomings. Also, the existing surveys/reviews are not comprehensive enough and are insufficient to highlight the challenges in the FSM domain along with their possible solutions.  Consequently, there is a need for a survey that incorporates recent techniques. Therefore, this study aimed to comprehensively survey the current research in the field of FSM. In this survey the key characteristics of each FSM approach are analyzed, such as the proposed methodology, which type of graph structure is used, applied similarity measures, metrics used for measuring the performance, uncertainty handled or not, used data set, capabilities of the techniques, evidence used and limitations of these techniques. As a result, this paper identifies the current status of the research in the FSM, and future research directions in this field are determined based on opportunities and several open issues in FSM domination. These research directions, facilitate the exploration of the domain and the development of optimal techniques to address FSM.

The linear Black-Scholes-Merton options pricing model has been (and is still) used since 1973 to attempt to estimate the option price for an underlying asset. In the last thirty years, researchers have introduced nonlinear feedback into this model to handle transaction costs, illiquidity, fluctuating markets and other market conditions. In this paper, we show that the Power Series Method (PSM) can be used to obtain accurate and straightforward results for certain nonlinear terms for several practical situations. From the presentation, one can ascertain how to modify the results demonstrated here for other types of nonlinear feedback.  Risk management applications using comparative statics are natural extensions of the PSM framework. Both numeric and symbolic solutions using PSM are presented.

This chapter aims to elucidate the concepts of non-informative and informative prior distributions concerning the variance, a crucial component of the unknown parameter within a normal distribution. The variance serves as a representation of the distribution's spread or variability. The estimation variance is exhibited in the point and interval estimations based on non-informative and informative prior distributions. Point estimation entails furnishing a specific value to estimate a population parameter. In contrast, interval estimation provides a range or interval of values to estimate a population parameter, commonly called a confidence interval. While non-informative priors express a need for prior knowledge, informative priors bring valuable insight into the modeling process. The non-informative priors encompass the maximum likelihood method, the Jackknife method, and the bootstrap method. Maximum likelihood is widely recognized for approximating parameters and boasting properties such as unbiased estimation, consistency, and efficiency. The informative prior distribution employs the Bayesian and Markov Chain Monte Carlo methods. These methods involve the prior distribution, given the probability distribution and approach to the posterior distribution.

Data aggregation represents a fundamental process within wireless sensor networks, facilitating the transmission of environmental data to end-users via base stations. Despite its critical role, data aggregation often receives less attention compared to routing and energy optimization challenges. This work presents a comprehensive review of existing data aggregation schemes, with a specific emphasis on correlational-based approaches. Our analysis reveals a significant gap in research dedicated to correlational-based data aggregation techniques. Furthermore, existing methods tend to overlook crucial factors such as data quality, computational complexity, and appropriate benchmarking. Addressing these unresolved issues is essential for enhancing the reliability and quality of data aggregation processes in wireless sensor networks. This chapter outlines the key challenges and opportunities for future investigations in this domain.

Data depth concept used to measure the deepness of a given point in the entire multivariate data cloud. It leads to center-outward ordering of sample points used rather than usual smallest to largest rank. The ordering starts from middle and moves in all directions. Multivariate location and scatter can be computed by using the depth value of each data point. Various depth procedures have been established by many authors. In this paper, a new depth procedure is proposed, namely Modified Mahalanobis Depth (MMD), which calculates depth based on robust distance with Minimum Covariance Determinant (MCD) approach and a weight function is established to determine the location and scale. The superiority of the proposed depth based procedure over existing depth procedures has been studied in simulated environment using R software with respect to application in discriminant analysis. In order to study the superiority of the proposed data depth procedure (MMD) it has been applied in discriminant analysis by comparing the Apparent Error Rate (AER) in the context of classification problems. From the experiment, through real and simulation studies, it reveals that, the AER of proposed data depth procedure is almost similar to existing depth procedures in case of less contamination level. But, when the contamination level, sample size and the number of dimension increases, the AER of the proposed data depth procedure (MMD) is less compared with other existing depth procedures. The proposed depth procedure performs well when compared with the existing procedures even with higher contamination levels and larger sample sizes. Further, it is concluded that the proposed procedure gives more accuracy in the context of classifying the objects when compared with the existing procedures. The proposed procedure is most suitable to the research communities who are performing statistical data analysis techniques by computing the measure of location and scatter. The data depth procedure introduced in this thesis can be beneficial to researchers, who work on machine learning techniques by considering the factors such as noise, computational time, ease algorithm approach and high dimensionality.

This paper examines the emerging need for the forensic investigation of artificial intelligence (AI) systems. As AI-based tools and algorithms become increasingly integrated into high-stakes domains like law enforcement and cybersecurity, critical challenges arise in forensically auditing these technologies to ensure reliability, accountability, and transparency. This paper extensively discusses the ability of AI technologies, such as machine learning, to automate analyses, data extraction, and evidence classification efficiency. However, there are many issues in the forensic analysis of AI systems, such as transparency in AI decision-making processes, possible biases, adversarial threats, and vulnerabilities that can interrupt investigations. This study employs a systematic literature review methodology for lately published literature, screening and incorporating 15 sources that are evaluated in this emerging area of AI in forensics. Themes obtained from the sources by thematic analysis are pertinent to both current AI applications in forensic investigation and ongoing impediments in AI system evaluation. Practice implications include the requirement for more robust, dependable AI for accountability and formalized ethical and legal frameworks to make sure that algorithmically produced evidence is reliable and admissible. Overall, taking into account the obtained knowledge, more interdisciplinary research is crucial for the successful integration of AI and forensic science.

In this paper we consider the forced KdV equation as a model for Tsunami generation. We derive analytical and approximate analytical solutions for different forces. Numerical simulation is performed. We make use of the Method of Line and a new \(\varphi\)-splines method.

In the present investigation, some novel analytical approximations to both unforced and forced pendulum-cart system oscillators are obtained. In our investigation, the ansatz method is employed for getting the approximations for the mentioned system. The obtained approximations are compared with the Runge-Kutta (RK4) numerical approximation. Also, the global maximum error is estimated as compared to RK4 numerical approximation.