A new class of orthogonal polynomials is introduced which generalizes the Bernstein-Szegö polynomials and includes the associated polynomials as well. The purpose of this paper is to give a natural extension of the Bernstein-Szegö orthogonal polynomials for a general class of weight functions. A nonnegative function w defined on the real line is called a weight function if w > 0, fRw > 0 and al...

A polynomial curve of degree 5, α, is a helix if and only if both ||α′|| and ||α′ ∧ α′′|| are polynomial functions.

In this paper we first characterize all quasi–homogeneous but non–homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.

Define a covering system (or covering) of the integers as a finite collection of congruences x ≡ aj (mod mj), with 1 ≤ j ≤ r, such that every integer satisfies at least one of these congruences. As an interesting application of coverings, W. Sierpiński [4] showed that there are odd positive integers k for which k · 2 + 1 is composite for all integers n ≥ 0. For d ∈ Z, the first author [1] consi...

In this paper, using a method of Luca and the author, we find all values x such that the quadratic polynomials x + 1, x + 4, x + 2 and x − 2 are 200-smooth and all values x such that the quadratic polynomial x − 4 is 100-smooth.

The partition function of the Potts model can be expressed in terms of famous combinatorial graph polynomials for restricted Hamiltonians. We investigate the Hamiltonian with variable edge interaction and external magnetic field, for which the partition function is the V-polynomial. Because the partition function is a polynomial, we can use motivic methods from algebraic geometry to characteriz...