Symmetries and similarities of planar algebraic curves using harmonic polynomials

Symmetries and similarities of planar algebraic curves using harmonic polynomials

We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a similarity transformation. Both algorithms are based on the fact, well-known in Harmonic Analysis, that the Laplacian operator commutes with orthogonal transformatio...

Algebraic Curves, Symmetries and Satellites.

Harmonic Deformation of Planar Curves

Chromatic Harmonic Indices and Chromatic Harmonic Polynomials of Certain Graphs

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

Planar Harmonic Polynomials of Type B

The hyperoctahedral group acting on R is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators {Ti : 1 ≤ i ≤ N}. These operators are analogous to partial derivative operators. This paper finds all the polynomials h on R which are harmonic, ∆Bh = 0 and annihilated by Ti for i > 2, where the Laplacian ∆B = ∑ N i=1 T 2 i . They are given explic...