We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x = T3(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials and b + degC = 3N . If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a ≤ 3. Most results are derived from continued fractions and their matrix represe...

We show that every rational knot K of crossing number N admits a polynomial parametrization x = Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a = 3 and b + degC = 3N. We show that every rational knot also admits a polynomial parametrization with a = 4. If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic k...

Precision-Recall analysis abounds in applications of binary classification where true negatives do not add value and hence should not affect assessment of the classifier’s performance. Perhaps inspired by the many advantages of receiver operating characteristic (ROC) curves and the area under such curves for accuracybased performance assessment, many researchers have taken to report PrecisionRe...

We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and Green's interior where the boundary surfaces are smooth convex. Both inequalities involve integrals over mean curvature Gaussian on an equipotential surface, normal derivative of potential thereupon. These generalize a conservation law curves dimension two, offer solutions to free p...

We have developed an alternate method of representing harmonic amplitude envelopes of musical instrument sounds using principal component analysis. Statistical analysis reveals considerable correlation between the harmonic amplitude values at different time positions in the envelopes. This correlation is exploited in order to reduce the dimensionality of envelope specification. It was found tha...