In this paper, we introduce the concept of fuzzy bipolar soft quasi-ideals in ordered semigroup theory. First some characteristics of the structure are examined and hence a few useful results are established. It is proved, among others, that the concepts of fuzzy bipolar soft bi-ideal and fuzzy bipolar soft quasi-ideal in regular ordered semigroups coincide. In addition, fuzzy bipolar soft quasi-ideals over ordered semigroups are linked with the ordinary quasi-ideals. Thereafter, a few classes of ordered semigroups are characterized in terms of their fuzzy bipolar soft left, fuzzy bipolar soft right and fuzzy bipolar soft quasi-ideals, and thus some important characterization theorems are established. We also define fuzzy bipolar soft semiprime quasi-ideals and characterize completely regular ordered semigroups by their fuzzy bipolar soft (semiprime) quasi-ideals. It is proved that an ordered semigroup S is completely regular if and only if every fuzzy bipolar soft quasi-ideal λA over S is a fuzzy bipolar soft semiprime quasi-ideal.

In this paper, we formulate the degree-based topological indices such as first Zagreb index, second Zagreb index, third Zagreb index, atom bond connectivity index, geometric-arithmetic (GA) index, general sum connectivity index, hyper-Zagreb index, augmented Zagreb index, sum connectivity index and general Randic index Rλ (℘), for λ = © −1, 1, − 1 2 , 1 2 ª for grape seed Proanthocyanidins network. By using above-mentioned topological indices, this study provides us different outcomes.

We establish the structure of the centre of V∗(F2 k (M2n+1 )), 1 V∗(F2 k (M2n+1 × C2)) and V∗(F2 k ((M2n+1 × C2) × C2)) over a finite field of characteristics 2 where M2n+1 =< ψ, λ|ψ 2n = λ 2 = 1, λψ = ψ 2n+1λ > is the Modular group having order 2 n+1 and C2 is a cyclic group of order 2

: In this paper, we produce a novel framework of a subclass of convex functions that is exponentially convex functions. Moreover, it is observed that the new concept helps to build new inequalities of Petrovic’s ´ type by employing exponentially convex functions. We also introduce the idea of coordinated exponentially convex functions and derive Petrovic’s ´ type inequality for coordinated exponentially convex functions. We also find Lagrange type and Cauchy type mean value theorems for Petrovic’s ´ type inequality for exponentially convex and coordinated exponentially convex functions. Our consequences with this new generalizations has the abilities to be implemented for the evaluation of many mathematical problems.

Let p be a prime greater than or equal to 5. In this paper, by using the harmonic numbers and Fermat quotient we establish congruences involving the sums p−1 X2 k=1 µ k r ¶ Hk, p−1 X2 k=1 ¡ 2k k ¢2 16k H (2) k and p−1 X2 k=1 1 4 k µ 2k k ¶ H (3) k . For example, p−1 X2 k=0 ¡ 2k k ¢2 16k H (2) k ≡ 4E2p−4 − 8Ep−3 ¡ mod p 2 ¢ , where H (m) k are the generalized harmonic numbers of order m and En are Euler numbers

Soft set theory is considered as one of the best effective tool which provides parameterization approach to tackle the inadequacy of fuzzy set. So far, it has been applied to different mathematical concepts such as set operations, algebraic structure (e.g., group and ring theory) and topological spaces. Many researchers have studied classical concept of convex and concave set under fuzzy-like, soft-like and fuzzy soft-like environments. In this paper, new notions of (m, n)-convex and (m, n)- concave fuzzy soft sets are developed first and then their versions for first and second senses are established. Further some known classical results and properties are generalized under fuzzy soft set environment. Moreover, special cases of (m, n)-convexity on fuzzy soft sets are established

The purpose of this work is to propose generalized extensions of morphisms in the geometry of Grassmannian configuration and Goncharov motivic chain complexes. This work has two major divisions: in its first part the geometry of these complexes will be extended for weight n = 6, secondly, the generalization of this extension for any weight N will be presented. The generalized commutative diagram will also be exhibited