On maximal operators on $k$-spheres in $\mathbb {Z}^{n}$

On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

We study approximation of Boolean functions by low-degree polynomials over the ring Z/2kZ. More precisely, given a Boolean function F : {0, 1}n → {0, 1}, define its k-lift to be Fk : {0, 1}n → {0, 2k−1} by Fk(x) = 2k−F(x) (mod 2k). We consider the fractional agreement (which we refer to as γd,k(F)) of Fk with degree d polynomials from Z/2 Z[x1, . . . , xn]. Our results are the following: • Incr...

On k-stellated and k-stacked spheres

We introduce the class Σk(d) of k-stellated (combinatorial) spheres of dimension d (0 ≤ k ≤ d+1) and compare and contrast it with the class Sk(d) (0 ≤ k ≤ d) of k-stacked homology d-spheres. We have Σ1(d) = S1(d), and Σk(d) ⊆ Sk(d) for d ≥ 2k− 1. However, for each k ≥ 2 there are k-stacked spheres which are not k-stellated. For d ≤ 2k− 2, the existence of k-stellated spheres which are not k-sta...

Dirac Operators on Quantum Two Spheres

We investigate the spin 1/2 fermions on quantum two spheres. It is shown that the wave functions of fermions and a Dirac Operator on quantum two spheres can be constructed in a manifestly covariant way under the quantum group SU(2) q. The concept of total angular momentum and chirality can be expressed by using q-analog of Pauli-matrices and appropriate commutation relations.

On the Maximal Degree of the K-Step Obrechkoff’s Method

On k-Maximal Strength Digraphs

Let k > 0 be an integer and let D be a simple digraph on n > k vertices. We prove that If |A(D)| > k(2n − k − 1)+ ( n − k 2 )