In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform al...

The fundamental matrix solution of the quantum Knizhnik-Zamolodchikov equation associated with Uq(ŝl2) is constructed for |q| = 1. The formula for its determinant is given in terms of the double sine function.

Two integral solutions of q-difference equations of the hypergeometric type with |q| = 1 are constructed by using the double sine function. One is an integral of the Barnes type and the other is of the Euler type.

We have presented a new hyperbolic auxiliary function method for obtaining traveling wave solutions of nonlinear partial differential equations. Applying this, exact traveling wave solutions for the coupled Sine-Gordon equations are constructed.

Motivated by models in engineering and also biology we determine closed form the probability density function of angle shaped two random chords a fixed disc. Our main result focus on event which intersection locates inside disc establishes sine law.

The generalized hypergeometric function has three singular points 0, 1, and ∞. Connection problem between fundamental solutions at 0 ∞ was completely solved. However, connection 1 been not solved because of holomorphic 1. In this paper, we solve for the function, using analytic continuation integral representation. Namely, all coefficients are products sine cosecant. other words, give a good sy...

In this paper we evaluate Euler-like sums involving harmonic/odd harmonic numbers and central binomial coefficients by using the expansion for powers of inverse sine function. We also a number integrals which in turn give rise to new coefficient identities.

In this work, a microcontroller based circuit that performs the functions of an inverter /AVR hardware essentially consists of a PIC 16F877A microcontroller based circuit,. SG 3524 based modified sine-wave inverter driver circuit, a step-up. Modified sine wave inverter is designed to using pic microcontroller and push pull topology. output of control circuit for modified sine wave inverter The ...

We analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry. We additionally obtain new representations of the Clausen function Cl2 and the Catalan constant G = Cl2(π/2), as well as new relations between sine and Clausen function values.