Journal of AppliedMath

This paper is a first of the series of three papers which provide a general proof to validate the Goldbach conjecture. This conjecture states that every even number can be expressed as a summation of two prime numbers. At the onset, concept of successive-addition of‐digits‐of‐an‐integer‐number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction are discussed. SADN classifies odd numbers into 3 sequences—the S1, S3 and S5 sequences—which comprise of odd numbers having SADN 7 or 4 or 1; SADN 3 or 9 or 6 and SADN 5 or 2 or 8 respectively. The S1 and S5 sequences are of interest in the analysis. Furthermore, composites on the S1 sequence are derived as products of intra-sequence elements of the S1 and S5 sequences while composites on the S5 sequence are derived as products of inter-sequence elements of the S1 and S5 sequences. SADN also shows why such combinations for even numbers of SADN (1, 4, 7) will be found on the S5 sequence while those for even numbers of SADN (2, 5, 8) will lie on the S1 sequence and both the sequences have a role to play in identifying the prime number combinations for even numbers with SADN (3, 6, 9). Thereafter, analysis moves to calculating the total number of combinations for a given even number that would include combinations in the nature of two composites (c1 + c2); prime & composite (p + c) and two primes (p1 + p2). Identifying the total number of c1 + c2 and p + c combinations yield the number of p1 + p2 combinations. The logic employed in present discussion shows that at least one such p1 + p2 combination exists for the even numbers having SADN digit within 1 to 9. This encompasses all even numbers and hence generalizes this method for all even numbers.

The intuitionistic fuzzy sets, in which the elements of the universe have their membership and non-membership degrees in [0, 1], is a generalization of Zadeh’s fuzzy set. In this paper intuitionistic fuzzy sets are used as tools for assessment and decision making. This is useful in cases where one is not sure about the suitability of the linguistic characterizations assigned to each element of the universal set. Further, it is described how the notions of convergence, continuity, compactness, and of Hausdorff topological space are extended to intuitionistic fuzzy topological spaces. Applications illustrating our results are also presented.

A Halin graph is a graph constructed by embedding a tree with no vertex of degree two in the plane and then adding a cycle to join the tree’s leaves. The Halin Turán number of a graph F , denoted as exH(n, F ), is the maximum number of edges in an n-vertex Halin graph. In this paper, we give the exact value of exH(n, C4), where C4 is a cycle of length 4. We also pose a conjecture for the Halin Turán number of longer cycles.

The current work looks at certain geometric requirements that must be satisfied for an invariant submanifold of an almost α- cosymplectic (k, µ, ν)-manifolds to be totally geodesic. Consequently, we obtain some interesting results invariant submanifolds of an almost cosymplectic (k, µ, ν)-manifolds. Additionally, we give an example on 5-dimensional case.

Intrusion detection in information technology as well as operational technology networks is highly required in modern day systems due to the increased spate of cyber-attacks in both number and complexity. Anomaly-based intrusion detection systems which have the capacity to detect novel or zero-day attacks are highly employed in this regard. One important component of anomaly-based intrusion detection systems which ensures their behaviour is artificial intelligence in general and machine learning in particular. The burden in modern day cybersecurity research is to investigate and develop models that can outperform existing ones. This paper is aimed at developing a hybrid decision tree model using the stacking ensemble approach. Performances were measured on the basis of recall, precision, accuracy, F1-score, receiver operating characteristics and confusion matrices. The hybrid model presented a precision of 97%, accuracy of 81%, F1-score of 80% and AUC score of 0.96, respectively.

This paper deals with the various definitions involved in the very old yet novel topic called fractional calculus. This survey intends to report some of the major works carried out in the arena of fractional calculus that took place since 2010. Fractional calculus is a prominent topic for research within the discipline of applied mathematics doe to its usefulness in solving problems in several different branches of science, engineering, medicine, finance, economics and the likes. With the various definitions involved in this field, we explore the various models taken into consideration to study the effect and impact of fractional calculus to understand how the dynamics of such models change.