The unit vectors transformation between the Cartesian and novel similar oblate spheroidal coordinates, vice versa, is derived. It facilitates of vector fields these two types orthogonal coordinates can advantageously simplify solutions problems exhibiting geometry. Several examples demonstrate use derived relations. Generalized sine cosine applicable in coordinate system are introduced as well.

We investigate bi–Hamiltonian structures and related mKdV hierarchy of solitonic equations generated by (semi) Riemannian metrics and curve flow of non–stretching curves. The corresponding nonholonomic tangent space geometry is defined by canonically induced nonlinear connections, Sasaki type metrics and linear connections. One yields couples of generalized sine–Gordon equations when the corres...

In this paper, with the aid of Maple, the (G ′ /G) and (1/G ′ )-expansion methods are applied to determine the exact solutions of (N+1)-dimensional generalized Boussinesq equation and (N+1)-dimensional sine-cosine-Gordon equation.These equations play a very important role in mathematical physics and engineering sciences. This methods are more powerful and will be used in further works to establ...

We classify all twist-even squeezed states in string field theory which are diagonal in the κ-basis and simultaneously surface states. For this purpose, we derive a consistency condition for the maps defining κ-diagonal surface states. It restricts these maps to a two-parameter family of Jacobi sine functions. Not all of them are admissible maps for surface states; standard requirements single ...

In this paper, we discuss the Hamiltonian structure of Korteweg–de Vries equation, modified Korteweg–de Vries equation, and generalized Korteweg– de Vries equation. We proposed the Sine-function algorithm to obtain the exact solution for non-linear partial differential equations. This method is used to obtain the exact solutions for KdV, mKdV and GKdV equations. Also, we have applied the method...

The purpose of this paper is to present a survey of recent developments concerning the distributions of occupation times of Brownian motion and their applications in mathematical finance. The main result is a closed form version for Akahori’s generalized arc-sine law which can be exploited for pricing some innovative types of options in the Black & Scholes model. Moreover a straightforward proo...

A general approach to arithmetic Fourier transforms is developed. The implementation is based on sine and cosine killer procedures pertaining to a generalized Möbius transform involving reduced periodic multiplicative arithmetical functions. It is shown that cosine killer procedures exist whenever one half of Euler’s totient function of the order of the transform is odd. Primitive roots and ind...