in this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. we obtain one results, which can give a negative answer to an uniqueness question relate to the bruck conjecture dealt by liu and yang. meanwhile, we also establish a difference analogue of the bruck conjecture for entire functions of order less than 2, ...

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ+f(x)ẋ+g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m− 1)/2] limit cycles, where [·] denotes the integer part function.

Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0, 1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear. We focus on the problem of extending partial representations...

Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence X , an integer k ≥ −1 and an upper set in the lattice of flats of the matroid defined by X , we define and study the associated hierarchical zonotopal power ideal. This ideal is generated by powers ...

Let (a0, · · · , ak−1) be a sequence of positive integers and m a positive integer. We prove that “almost every” real quadratic unit ǫ of norm (−1)k admits at leastm distinct factorizations into a product of purely periodic irrationals of the form ǫ = [a0; a1, . . . , ak−1, x, y]× [a1; a2, . . . , x, y, a0] × · · · × [y; a0, . . . , ak−1, x]. Periods exhibited in this expression are not assumed...