Research Updates in Mathematics and Computer Science Vol. 7

Let \(p(z)\) be a polynomial of degree \(n\) having no zero in \(|z|<1\), then Erdös conjectured and later Lax [Bull. Amer. Math. Soc., \(50(1944), 509-513\) ] prove that
\[
\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{2} \max _{|z|=1}|p(z)|
\]

This Erdös-Lax's inequality was generalized for the first time by Malik [J. London Math. Soc., 1(1969), 57-60] that if \(p(z)\) is a polynomial of degree \(n\) having no zero in \(|z|<k, k \geq 1\), then
\[
\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{1+k} \max _{|z|=1}|p(z)|
\]

For the class of polynomials not vanishing in \(|z|<k, k \leq 1\), the precise estimate for maximum of \(\left|p^{\prime}(z)\right|\) on \(|z|=1\), in general, does not seem to be easily obtainable. But for the particular class of polynomials having all its zeros on \(|z|=k, k \leq 1\), Govil [J. Math. and Phy. Sci., 14(1980), 183-187] was able to prove that
\[
\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{k^{n-1}+k^n} \max _{|z|=1}|p(z)| .
\]

In this article, we compare some inequalities of later type concerning the ordinary and polar derivatives of the polynomial.

This chapter is an introduction to a new type of analogue named Q -analogue for some operators. Here we have considered well-known operators named Baskakov Durrmeyer operators. This new type of analogue is considered as reverse order of q-analogue. In this chapter, we establish a direct approximation theorem, a weighted approximation theorem followed by the estimations of the rate of convergence of these new types of operators for functions of polynomial growth on the interval [0,\(\infty\)).

The \(\\{M}|G| \infty\)  queue system transient probabilities, with time origin at the beginning of a busy period, are determined. It is highlighted that the obtained distribution mean and variance study as time functions. In this study, it is a determinant of the service time hazard rate function and two induced differential equations. The study model has potential applications in various real-world scenarios, such as epidemics and unemployment. The model's application in these areas offers insights into potential mitigating measures, including vaccination and medical care in epidemics, and government support and training in unemployment. Also, we will show how the results obtained can be applied in modeling epidemic and unemployment situations.

A major driver in the rethinking of sustainable engineering and management practices is artificial intelligence (AI), especially in the use of cutting-edge deep learning techniques. The primary focus of this study is to assess the efficacy of Long Short-Term Memory (LSTM) networks as an example of the substantial role that artificial intelligence (AI) plays. When it comes to improving operational performance, optimizing resource allocation, and reducing environmental implications, the Long Short-Term Memory (LSTM) model—a complex sort of recurrent neural network—is crucial. An interesting case study illustrating the use of LSTM algorithms to optimize smart building energy usage in real time is included in this research. By utilizing LSTM for comprehensive pattern analysis and making real-time adjustments, AI exhibits impressive efficiency gains in reducing energy waste. Within the broader context of sustainable engineering, this study demonstrates the effectiveness and efficiency of Long Short-Term Memory (LSTM), therefore contributing significantly to the development of a resilient and ecologically conscious future.

The next generation of wireless communication 5G, is focusing on advanced antenna technologies like massive MIMO, phased arrays, and mm-wave bands to obtain a high data rate of 10Gbps. To meet the high data rate, the mm-wave band stands promising. The mmWave communication has one major drawback which is, high signal loss. To overcome this disadvantage, there is a need to design a high-gain antenna. To meet the required data rate, a MIMO antenna must be incorporated into the system. In this paper initially, a single antenna is designed to operate at 28GHz. Later, an antenna array of 4 elements is constructed. The antenna array is excited using corporate feed techniques. It is observed that the gain enhancement of 14dB is achieved with the antenna array. The designed 4-element antenna array is transformed into a 4-port MIMO system. The performance criteria to evaluate the MIMO system is the Envelope correlation coefficient. The obtained ECC is around 0.05. The design also focuses on the radiation pattern stability in the operating bandwidth. The array operates from 27.9GHz to 28.4GHz with 900MHz impedance bandwidth. The radiation pattern obtained when all the ports are excited with the same amplitude and phase is stable for the operating frequency band. With this stability, the designed antenna can be used in beam steering applications. The half-power beamwidth of the array is 11.1 degrees.

We seek to unearth which distribution best describes traffic behaviour to help solve the problem of congestion on campus from a mathematical point of view. Telecommunications traffic engineering (i.e. Tele -traffic engineering) is the application of traffic engineering theory to telecommunications. Tele-traffic engineers use their cognition of statistics including queuing theory, the nature of traffic, their practical fashion model, and their mensuration and computer simulation to make predictions and to program telecommunication networks such as a telephone network or the Internet. Internet traffic is a flow of data across the internet. This flow exhibits certain behaviour in accordance with a Probability distribution.  Statistical analysis was performed to understand the characteristics of the traffic population. The empirical cumulative distribution process (CDF) together with essential statistical parameters were benchmarked. The goodness of Fit (GOF) test using the Anderson-Darling (AD) estimation method was applied to establish the best probability distribution model which describes the situation alongside with Probability plot. Deep-down opinion or analysis hints that if the period of time (or space) of grouping is increased well enough, the degree of data relationships will eventually become unimportant by scaling. The N-Flow traffic model was used to determine the burstiness of internet traffic in finite sessions. The Empirical and Theoretical CDF graph was used to determine the discrepancies between them. In the end, it was discovered that the data best fit normal distribution. From experimental and theoretical analysis, it is clear that the internet traffic behaviour in Ghana Technology University College is normally distributed.

Does the MAC address have any functions in the Public network? A Media Access Control Address (MAC Address) is a unique number assigned to a Network interface controller (NIC) card of a Personal Computer (PC) or Server or Router or Firewall or any other devices attached to Private or Public wired Network segments. MAC address indicates only the device's LAN or NIC card and the NIC card primarily holds the Internet Protocol (IP) address [1]. This paper differentiates each function of MAC address (not MAC card or LAN card, LAN card is essential for Networking) and IP address and describes their scopes of work and limitations.

This chapter presents a VAR analysis framework for pass-through effects emanating from macroeconomic shocks. The framework involves two critical steps. The first step involves estimating a VAR model parameter that helps in establishing the causal link among the variables. The estimated VAR model is utilized in structural analysis to determine the behavior of a variable in response to a shock given the causal link. In the structural analysis step; Granger causality, impulse response function and forecast error variance decomposition are considered. The framework is applied to analyze exchange rate pass-through in Kenya.

The research provides the intertwined realms of Data Lineage and Data Governance, two crucial facets of contemporary data management within organizations. Collectively, they ensure data quality, security, compliance, and transparency, all of which are essential for informed decision-making. Data Lineage, as the flow and transformation of data through pipelines, is explored in tandem with Data Governance, which provides the principles and frameworks for effective data management. Data lineage assures data accountability by showing who is responsible for various data sets and how data is managed across different processes and systems. The article elucidates the prerequisites for successful data governance, highlighting the pivotal role played by executive support, clear business objectives, comprehensive data inventories, and robust security measures. The research further discusses how Data Lineage aligns with the fundamental principles of Data Governance, including data transparency, accountability, quality assurance, security, and regulatory compliance. By tracing the origin and evolution of data, Data Lineage ensures that data is trustworthy and can be relied upon for informed decision-making. The research suggests that data lineage complements the applicability of data governance. The objectives required from data lineage must align with the data governance principles developed by individuals.  

In this scholarly article, we delve into the intricate realm of exponential Diophantine equations, specifically those pertaining to the reversal of digits denoted by \(29^x+92^y=z^2, \quad 38^x+83^y=z^2, \quad 47^x+74^y=z^2, \quad 56^x+65^y=z^2\). Within this context, to determine if a distinct solution exists for the considered exponential equation, it is finally revealed that it has infinitely many solutions, but all of the aforesaid exponential equations have a similar solution, identified as \((1,1,11)\).

Despite the elaborate Wiles demonstration, Fermat’s last theorem still attracts researchers to this aspect of number theory. This study has been structured in two parts. The first one describes the Minkowski natural space basis, where the discussion of Fermat’s theorem and the extensions to higher dimensional spacesss. In the second, many examples of N-dimensional vectors obeying a Fermat-like rule for various powers are presented and discussed. The Fermat last theorem, defined in (2+1)-dimensional Minkowski spaces, is discussed and extended in natural and rational Mikowski’s spaces. Several pieces of computational interest are given, with many practical examples. A definition of Fermat vector order, Fermat surfaces, and Fermat surface radius is given. Several conjectures are discussed, among them the existence of a Fermat theorem in (3+1)-dimensional Minkowski spaces.

The primary objective of the cluster analysis is to form clusters such that each cluster is an homogeneous as possible with respect to items of interest and as different as possible between clusters. Cluster analysis is a mathematical technique in Multivariate Data Analysis which indicates the proper guidelines in grouping the data into clusters. Factor Analysis is a combination of intrinsic decomposition and recurrent factor decomposition [1]. It is surpassing Statistical approaches, factor decomposition has endured from incertitude review justification. We can understand the concept with illustrated notations of cluster Analysis and various Clustering Techniques in this Research paper. Similarity and Dissimilarity measures and Dendrogram Analysis will be computed as required measures for Analysis. The approach of factor analysis is helpful in figuring out the underlying, hidden factors that underlie the correlations between the variables. Sometimes it's crucial to identify and isolate these facts for use in a variety of statistical techniques across different domains. With the help of illustrated factor analysis methodologies, we may comprehend the significance of factor analysis and its key concepts. We can estimate the Basic Factor Modeling and Factor Loadings, and also Factor Rotation process. Provides the complete application process and approaches of Principal Factor M.L. Factor and PCA comparison of Factor Analysis in this Research paper Since all these methods use some sort of similarity measures. The same thing can be carried out with a similarity measure matrix also. In the behavioral and social sciences, researchers need to develop scales for the various un-observable factors such as attitudes, images, intelligence, personality and patriotism.

This paper elucidates the connections between intuitionistic \(\beta\) B - open sets, intuitionistic \(\beta\) B - interior, intuitionistic \(\beta\) B - closure, intuitionistic \(\beta\) AB - open sets, and intuitionistic \(\beta\) C - open sets.