Problem statement: Frequent itemset mining is an important task in data mining to discover the hidden, interesting associations between items in the database based on the user-specified support and confidence thresholds. Approach: In order to find important associations, an appropriate support threshold has to be specified. The support threshold plays a key role in deciding the interesting itemsets. The rare itemsets may not found if a high threshold is set. Some uninteresting itemsets may appear if a low threshold is set. Results: This study proposes an approach to obtain the frequent itemsets involving rare items by setting the support thresholds automatically. Experimental results show that this approach produces rare and frequent itemsets in sparse and dense datasets. According to T20I6D100K, 97.76% of the FIs are generators wherein Mushrooms 1.38% of the FIs are the generators. Conclusion: The proposed algorithm produces both frequent and rare itemsets in an effective way. In future, computational efforts can still be reduced by implementing the algorithm as parallel algorithm

This article is an attempt to study the class of impulsive vector partial functional differential equations with continuous distributed deviating arguments. The main aim of this paper is to present some sufficient conditions for the H-oscillation of solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. Our main results are point up with a suitable example.

In this article, we study the oscillatory behavior of fractional order Emden-Fowler qdifference equations of the form Dq r(t) c D α q z(t) + φ(t) |x(σ(t))| γ−1 x(σ(t)) = 0, t ≥ t0, where z(t) = x(t) + p(t)x(t − τ), c D α q denotes the Caputo q-fractional derivative of order α,0 < α ≤ 1. Using the generalized Riccati technique, new oscillation criteria are established.

In this paper, we discuss solvability of the Diophantine equation of the form (x1 + x2 + x3 + • • • + xn) 2 = x1x2x3• • •xn. An explicit closed form solutions are obtained by using the theory of Pell's equations which generate the different families of positive integral solutions to this equation. Some illustrative examples are inserted in order to explain the effectiveness of our results.

In this paper, we are concerned with the oscillation criteria for self-adjoint alpha-fractional matrix differential system with damping term. By using the generalized Riccati technique and the averaging technique, some new oscillation criteria are obtained.