In this paper we study the lower bounds of second-order nonlinearities of bent functions constructed by modifying certain cubic Maiorana-McFarland type bent functions.

We prove that cubic homogeneous bent functions f : V2n → GF(2) exist for all n ≥ 3 except for n = 4.

We consider a special class of two-dimensional discrete equations defined by relations on elementary N ×N squares, N > 2, of the square lattice Z, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary N × N squares, N > 2, in the cubic lattice Z. For an arbitrary N we prove such consistency on cubic lattices for tw...

We investigate the simultaneous existence of two centers (bi-center) for two families of planar Z2-equivariant differential systems. First we present necessary and sufficient conditions for the existence of an isochronous bi-center for a planar Z2-equivariant cubic system having two centers at the points (−1, 0) and (1, 0), completing the study done by Liu and Li (2011). Next, we give condition...

2 The reciprocity problem 3 2.1 Quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Cubic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Biquadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Hilber...

In this paper, we prove the generalized Hyres–Ulam–Rassias stability of the mixed type cubic and quartic functional equation f (x + 2y) + f (x − 2y) = 4(f (x + y) + f (x − y)) − 24f (y) − 6f (x) + 3f (2y) in non-Archimedean ℓ-fuzzy normed spaces.

In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation 3(f(x+ 2y) + f(x− 2y)) = 12(f(x + y) + f(x− y)) + 4f(3y)− 18f(2y) + 36f(y)− 18f(x).

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 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.

In this paper we completed the classification of the phase portraits in the Poincaré disc of uniform isochronous cubic and quartic centers previously studied by several authors. There are three and fourteen different topological phase portraits for the uniform isochronous cubic and quartic centers respectively.