In this paper, we obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation 4(f(3x + y) + f(3x− y)) = −12(f(x + y) + f(x− y)) + 12(f(2x + y) + f(2x− y))− 8f(y)− 192f(x) + f(2y) + 30f(2x).

We provide an elementary evaluation for the integral

For the family a0x 4 = a1y +a2z +a3v +a4w 4, a0, . . . , a4 > 0, of diagonal quartic threefolds, we study the behaviour of the height of the smallest rational point versus the Tamagawa type number introduced by E. Peyre.

We present an accurate method to compute the minimum distance between a point and an elliptical torus, which is called the orthogonal projection problem. The basic idea is to transform a geometric problem into finding the unique real solution of a quartic equation, which is fit for orthogonal projection of a point onto the elliptical torus. Firstly, we discuss the corresponding orthogonal proje...

We shall characterize the Fermat quartic K3 surface, among all K3 surfaces, by means of its finite group symmetries.

Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over Q. We show that X(Q) is non-empty provided that X(R) is non-empty and X has p-adic points for every prime p.

We prove generalized Hyres-Ulam-Rassias stability of the cubic functional equation f(kx + y) + f(kx − y) = k[f(x + y) + f(x − y)] + 2(k − k)f(x) for all k ∈ N and the quartic functional equation f(kx + y) + f(kx − y) = k[f(x + y) + f(x − y)] + 2k(k − 1)f(x)− 2(k − 1)f(y) for all k ∈ N in non-Archimedean normed spaces.

and Applied Analysis 3 In 2008, Gordji et al. 17 provided the solution as well as the stability of a mixed type cubic-quartic functional equation. We only mention here the papers 19, 32, 33 concerning the stability of the mixed type functional equations. In this paper, we deal with the following general cubic-quartic functional equation: f ( x ky ) f ( x − ky) k2(f(x y) f(x − y)) 2 ( 1 − k2 ) f...

In 1985, K. S. Williams, K. Hardy and C. Friesen [11] published a reciprocity formula that comprised all known rational quartic reciprocity laws. Their proof consisted in a long and complicated manipulation of Jacobi symbols and was subsequently simplified (and generalized) by R. Evans [3]. In this note we give a proof of their reciprocity law which is not only considerably shorter but also she...