In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n ≤ 5× 106 and n ≥ N = 1.191× 1019, with n, n+ 2, n2 + 4 square-free, the Thue equation Φn(x, y) = x 4 − nxy − (n + 2n + 4n+ 2)xy − nxy + y = 1 has no integral solution except the trivial ones: (1, 0), (−1, 0), (0, 1), (0,−1).

As it is well known, only a very limited number of one-dimensional potentials allow for an exact solution of the Schrödinger equation. This means that for many model potentials we must resort to numerical solution methods. For judging their accuracy, reliability, and speed, it is important to have high-precision values of certain nonexactly solvable potentials. The most investigated of such pot...

In this paper we first define homorooty between two integer numbers and study some of their properties. Thereafter we state some applications of the homorooty in studying and solving some Diophantine equations and systems, as an interesting and useful elementary method. Also by the homorooty, we state and prove the necessary and sufficient conditions for existence of finite solutions in a speci...

In this paper, we investigate the general solution and the generalized Hyers-Ulam stability of a new functional equation satisfied by $f(x) = x^{24}$, which is called quattuorvigintic functional equation in intuitionistic fuzzy normed spaces by using the fixed point method.These results can be regarded as an important extension of stability results corresponding to functional equations on norme...

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k + · · ·+xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we deter...

abstract. let x be a vector space over a field k of real or complex numbers.we will prove the superstability of the following golab-schinzel type equationf(x + g(x)y) = f(x)f(y); x; y 2 x;where f; g are unknown functions (satisfying some assumptions). then we generalize the superstability result for this equation with values in the field of complex numbers to the case of an arbitrary hilbert spac...

In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely Ḣ− 1 6 . The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao [19] we are able to work purely in the scaling space without additional regularity assumptions, allowing us to construct wave ope...

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k + · · ·+xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we deter...

We evaluate the shape and chirp of nanosecond pulses from a fiber laser passively mode locked with a nanotube-based saturable absorber by using a synchronously scanning streak camera and a monochromator to directly measure the pulse spectrogram. We show that the stable sech(2) output pulse possesses a predominantly linear chirp, with a residual quartic phase and low noise. Comparison with analy...

The bound state solutions of Schrodinger’s equation for the anharmonic oscillator potentials V = x2+ AxZk ( k = 2 , 3 , . . .) have been investigated, using elementary techniques of low-order variational perturbation theory. For the quartic oscillator ( k = 2) a scaled harmonic potential provides a remarkably accurate model for all A. Although this model is slightly less satisfactory for higher...