Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quaternion functions, we give a notion of an $\mathbb{H}$-derivative for functions of one quaternion variable. We show that the elementary quaternion functions introduced by Hamilton as well as the quaternion logarithm function possess such a derivative. We conclude by establishing rules for calculating $\mathbb{H}$-derivatives.

Families of Newton type methods for solving non-linear equations are obtained. These families consist of second and third order methods. Concrete two step methods are presented which are obtained as a unification of some existing methods in the literature and the standard secant method. It is shown that by this unification, the order of convergence of the methods increases. All the methods presented in this paper are well supported by examples.

The notion of capability for pairs of groups was defined by Ellis in 1996. In this paper, we extend the theory of $c$-capability for pairs of groups and introduce a criterion, denoted by $Z_c^*(G,N)$, for $c$-capability of a pair $(G,N)$ of groups. We also study the behavior of $Z_c^*(G,N)$ with respect to direct products of groups.

In this work, for the self similar sets satisfying the open set condition, we obtain an interesting character that the upper convex density at every similarly contractive fixed point equalling 1 implies that the upper convex densities at all the points are equal to 1. By using this result, we sufficiently answer the open problem 6 posed by Z. Zhou and L. Feng in [Twelve open problems on the exact value of the Hausdor measure and on topological entropy: A brief survey of recent results, Nonlinearity, 17(2004) 493–502.].

In this paper, we investigate the long-time behavior of the solutions for the Hirota equation with the periodic boundary condition. At first, by time uniform priori estimates of solutions, we obtain the existence of global solutions. Furthermore, we prove the existence of a global attractor. Finally, by squeezing property and Lipschitz continuity, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.

The objective of this paper is to obtain an upper bound to the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ for starlike and convex functions of order $\alpha$ $(0\leq\alpha\lt1)$, also for the inverse function of $f$, belonging to the class of convex functions of order $\alpha$, using Toeplitz determinants.