Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case ...

Assuming the Gowers Inverse conjecture and the Möbius conjecture for the finite parameter s, Green-Tao verified Dickson’s conjecture for lattices which are ranges of linear maps of complexity at most s. In this paper, we reformulate GreenTao’s theorem on Dickson’s conjecture, and prove that, if L is the range of a linear map of complexity s, and L1 is a sublattice of L of finite index, then L1 ...

Alexandr Kostochka In search of ways to attack Hadwiger’s Conjecture, Woodall and independently Seymour suggested to prove the weaker conjecture that Every (s + t)-chromatic graph has a Ks,t-minor. If the conjecture were true for all values of s and t, it would imply that for k ≥ 2 every (2k − 2)-chromatic graph has a Kk-minor. The conjecture is evident for s = 1. The validity of the conjecture...