Research Updates in Mathematics and Computer Science Vol. 2

A logic glitch is a kind of unwanted noise, its presence in the output signal can initiate an uncontrollable process, in the next level which is an input signal. There can be distinguished three types of noise that are introduced in (CLC) Combinational Logic Circuits, called hazards (Static, Dynamic and Function Hazards). In this paper, the authors continue the research that consists of a comparative study of two methods to eliminate the static hazard from logical functions, by using the form of POS – Product of Sums, static hazard “0”.

In the first method it’s used the consensus theorem to determine the cover term that is equal to the product of the two residual implicants, and in the second method, it’s resolved a Boolean equation system. The authors observed that in the second method, the digital hazard can be earlier detected. If the Boolean equation system is incompatible (doesn’t have solutions), the considered logical function doesn’t have the static 1 hazard regarding the coupled variable. Using logical computations, this method permits to determine the needed transitions to eliminate the digital hazard.

For many businesses, the quantity of sensitive information handled by external service providers is on the rise. With an expanding array of entities granted access to databases, the task of preventing and tracking data leaks becomes increasingly challenging. We address the problems associated with verifying ownership and unlawful distribution (leaking) of information in relational databases. Our suggested method provides a thorough strategy for identifying, discouraging, and tracking down data breaches in relational databases. Our primary descriptive use case is business process outsourcing scenarios, but our techniques can also be applied to other scenarios involving shared relational databases that need to maintain confidentiality and authenticity. Initially, fingerprinting and watermarking were employed to determine who owns what and keep track of illegally distributed numerical relational data. In our current work, we introduce "realistic but counterfeit" objects into our database, a technique akin to watermarking.

This paper focuses on the problem of using machine learning techniques on cancer/tumor prediction. One of the biggest applications of big data and machine learning is in the field of medical domain. Consequently, a health care organization that uses the techniques of machine learning and big data to treat patients see fewer mishaps or gets enough time to deal with them in advance. Classification and prediction of the images are fairly easy task for humans, but it takes more effort for a machine to do the same. Machine learning helps to attain this goal. It automates the task of classifying a large collection of images into different classes by labelling the incoming data and recognizes patterns in it, which is subsequently translated into valuable insights. Furthermore, the prediction of Osteosarcoma case for one of the four classes of tumor namely Non tumor, Non-Viable tumor, viable tumor, Viable: Non-Viable tumor has to be done. The quantitative analysis is done using various machine learning libraries of python. The three classification algorithms used for image analysis are random forest, SVM, and logistic regression. The metrics used for performing perspective analysis are precision, recall and F1 Score. The domain of medical imaging helps providing important information on anatomy and organ function subsequently detecting disease states. Although the characteristics of medical data make its analysis a big challenge notwithstanding that machine learning techniques could make the task easier. The results show that the random forest algorithm has performed best amongst the three classification algorithms when given with less complicated scenario, with prediction accuracy, precision, recall and f1 score of 100%. But the performance of every classification algorithm degrades when provided with the cases of Osteosarcoma which has got more complicated scatter graph. However, the logistic regression retains its performance by predicting tumor cases with 99% accuracy. For future scope, various other machine learning algorithms can be applied to observe their performance on the same set of features extracted from the cancer image data set.

This study introduces a novel high-precision quadrature rule, achieved by using two lower-precision quadrature rules. The focus is on facilitating the approximate evaluation of integrals over line segments in the complex plane, particularly for analytic functions. The versatility of the newly developed quadrature rule is demonstrated through its application to various mathematical scenarios. To assess the efficacy of the proposed quadrature rule, an asymptotic error estimate is provided. Numerical verification is then conducted to validate the accuracy and efficiency of the rule. The results from these numerical experiments highlight the superior precision of our quadrature rule when applied to the numerical integration of functions over complex line segments. This study significantly contributes to the advancement of numerical integration techniques, presenting a promising avenue for achieving heightened accuracy in the evaluation of integrals over complex domains, particularly in the context of analytic functions.

The purpose of this article is threefold:
(i) To prove that each real eigenvalue of \(\mathbf{A}=\mathbf{A}_R+i \mathbf{A}_I \in \mathbb{C}^{m \times m}\) is doubled in its real counterpart \(\mathbb{A}:=\left[\begin{array}{rr}\mathbf{A}_R & -\mathbf{A}_I \\ \mathbf{A}_I & \mathbf{A}_R\end{array}\right] \in \mathbb{R}^{2 m \times 2 m}\), whereas each complex eigenvalue of \(\mathbf{A}\) appears in \(\mathbb{A}\) together with its complex conjugate. Hence, if \(f(\lambda ; \mathbf{A}):=\operatorname{det}\left(\lambda \mathbf{I}_m-\mathbf{A}\right)=\lambda^m+\sum_{k=0}^{m-1} a_k \lambda^{m-k}\) is the characteristic equation of \(\mathbf{A}\) then \(f(\lambda ; \mathbb{A}):=\operatorname{det}\left(\lambda \mathbf{I}_{2 m}-\mathbb{A}\right)=\left(\lambda^m+\sum_{k=0}^{m-1} a_k \lambda^{m-k}\right)\left(\lambda^m+\sum_{k=0}^{m-1} \operatorname{conj}\left(a_k\right) \lambda^{m-k}\right)\) is the characteristic equation of \(\mathbb{A}\), where \(\mathbf{I}_m\) denotes the \(m\)-dimensional unit matrix and \(\operatorname{conj}(a)\) denotes \(a\) 's conjugate. Notice that the proof of this result was not trivial.
(ii) To give based on (i) another simple proof of the author's result that \(\operatorname{rank}(\mathbf{A})=\) \(r\) if and only if \(\operatorname{rank}(\mathbb{A})=2 r\), where \(\mathbf{A} \in \mathbb{C}^{m \times n}\) and \(\mathbb{A} \in \mathbb{R}^{2 m \times 2 n}\).
(iii) To improve Rump's lower bound on the minimal eigenvalue of an interval Hermitian matrix that also reduces its complexity by a factor of two and decreases the dimensionality of the chosen vertex matrices.

Data mining refers to a variety of techniques in fields of databases, machine learning and pattern recognition. The intent is to obtain useful patterns and associations from large collection of data.  The approach we consider is the use of attribute generalization using concept hierarchies. We will consider heterogeneous data with uncertainty; in particular we extend the previous techniques for fuzzy sets to data represented by intuitionistic or interval-valued fuzzy sets.

This chapter is concerned with the study of the construction of sequences of Diophantine triples ^(a,b,c) such that the product of any two elements of the set subtracted by a polynomial with integer coefficients is a perfect square and non-extendability of Gaussian Diophantine quadruples with suitable properties.

This work analyses the behavior, characteristics, flow development, and heat and mass transfer coefficient of nanofluids under laminar boundary layer flow over a horizontally inclined plate. A nanofluid is the suspension of ultrafine particles in a base fluid, which tremendously enhances the heat transfer characteristics of the original fluid. This work aims to compare the characteristics between two different nanofluids Silver and Aluminum Oxide with the same base fluid water under the heat and mass transfer nanofluid past a horizontally inclined plate. The Basic governing equations of the free convective flow for heat and mass transfer are converted to a non-dimensional partial differential equation using non-dimensional parameters and are solved numerically using crank Nicolson Implicit finite difference method. The final equations are solved using the MATLAB software and the results are shown as graphs. The velocity, temperature and concentration profiles are illustrated graphically to study about the various parameters Grashof Number, Angle of inclination, Nano particle volume fraction, Schmidt number and non-uniform surface temperature.

This study primarily focuses on skin cancer image segmentation based on symmetrical threshold contour algorithm Image segmentation is classification or identification of small patterns in the given images. Image segmentation is a fundamental step in image processing.  Image segmentation is the classification of an image into various groups. Research has been done in image segmentation using image clustering methods like k-means clustering algorithm. The approach describes the skin cancer image segmentation based on symmetrical threshold contour algorithm with similar thresholding values for segmentation of the accurate cancerous lesion. Skin cancer lesion shape and structure is the most important parameter in this method. In this paper, skin cancer image contour detection is based on symmetrical thresholding algorithm using MATLAB soft ware. Results of the proposed method are compared with dilation of image morphological algorithm and its contour. By observing the dilated image and its contour, the proposed method gives continuous and accurate contour.

Big data refers to any assortment of data which are outsized and intricate in nature such that conventional database administration systems and data processing tools cannot process. Feature selection serves primarily to reduce the processing burden of data mining models. To expedite the processing of large volumes of data, parallel processing is implemented using the MapReduce (MR) technique. MapReduce model is applied to big datasets, which is further divided into smaller partition. However, existing algorithms often fall short in enhancing classifier performance significantly. This research advocates for the use of the MR method to conduct feature selection in parallel, thereby improving performance. Additionally, to augment classifier efficacy, this study introduces an innovative approach combining Online Feature Selection (OFS) with an Accelerated Bat Algorithm (ABA) within a framework that pre-processes features ahead of time, without prior knowledge of the feature space. The proposed OFS-ABA method is designed to select relevant and non-redundant features within the MapReduce (MR) framework. Furthermore, an Ensemble Incremental Deep Multiple Layer Perceptron (EIDMLP) classifier is employed to classify dataset samples. The outputs from homogeneous IDMLP classifiers are aggregated using the EIDMPL classifier. The proposed feature selection method and classifier are extensively evaluated across three high-dimensional datasets. The results indicate that the MR-OFS-ABA method outperforms existing feature selection methods such as PSO, APSO, and ASAMO (Accelerated Simulated Annealing and Mutation Operator). Additionally, the performance of the EIDMLP classifier is compared with other existing classifiers including Naïve Bayes (NB), Hoeffding Tree (HT), and Fuzzy Minimal Consistent Class Subset Coverage (FMCCSC)-KNN (K Nearest Neighbor). The methodology is applied to three datasets, and the results are compared across four classifiers and three state-of-the-art feature selection algorithms. Overall, the findings of this research demonstrate improved accuracy and reduced processing time.

The aim of this paper, propose a mathematical modelling on two stage pre-emptive priority queuing system and minimizing the waiting time. Initially, system can be split into two units such as: ordinary unit and priority unit. Here each unit servers are not dependent and service time considered as K-phase Erlang distribution. According to this quickly assign the server for emergency customer without affecting the ordinary unit. In this case, derive the ’nth’ customer steady state probability in both units. A numerical example is included.