The purpose of this article is to give formulas for BlochKato’s exponential map and its dual for an absolutely crystalline padic representation V , in terms of the (φ,Γ)-module associated to V . As a corollary of these computations, we can give a very simple and slightly improved description of Perrin-Riou’s exponential map, which interpolates Bloch-Kato’s exponentials for the twists of V . Thi...

The Wiener index is the sum of distances between all pairs of vertices in a connected graph. In this paper, explicit expressions for the expected value of the Wiener index of three types of random pentagonal chains (cf. Figure 1) are obtained.

Abstract: In this paper, we demonstrate the effectiveness of the (G ′ G )-expansion method by seeking more exact solutions of the SRLW equation, the (2+1) dimensional PKP equation and the (3+1) dimensional potential-YTSF equation. By the method, the two nonlinear evolution equations are separately reduced to non-linear ordinary differential equations (ODE) by using a simple transformation. As a...

Recently, the Runge-Kutta methods, obtained via Taylor’s expansion is exist in the literature. In this study, we have derived explicit methods for problems of the form y′ = f(y) including second and third derivatives , by considering available Two-Derivative Runge-Kutta methods (TDRK). The methods use one evaluation of first derivative, one evaluation of second derivative and many evaluations o...