Combining results presented in two papers in this Monthly yields the following elementary result. Any line of best fit for the zeros of a polynomial is a line of best fit for its critical points. This note gives a generalization of results on cubic polynomials presented in [1]. Our notation will follow that paper. A line of best fit for a set of points in the plane is defined, as in [1, p. 682]...

Fitting an implicit polynomial (IP) to a data set usually suffers from the difficulty of determining a moderate polynomial degree. An over-low degree leads to inaccuracy than one expects, whereas an overhigh degree leads to global instability. We propose a method based on automatically determining the moderate degree in an incremental fitting process through using QR decomposition. This increme...

For the inverse synthetic aperture radar (ISAR) imaging of a target at a long range, range alignment using the existing polynomial method brings about poor results because the flight trajectory changes depending on the initial position, and the motion parameters, meaning the polynomial cannot fit the trajectory. This paper proposes an improved range alignment method that models the trajectory u...

Although algebraic or so-called “implicit polynomial” curves have been studied rather extensively for several decades, to the best of our knowledge, a dynamic formulation of them, similar to active contours, has not been done yet. This paper develops a dynamic formulation for implicit polynomial curves based on level set formalism. In particular, it is shown that utilization of an implicit poly...

Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improvin...

We investigate the separation of irreducible characters by blocks at different primes and the covering of irreducible characters by blocks (viewed as sets of characters); these notions are used to prove results on the group structure. The covering of all characters of a group by principal blocks is only possible when already one principal block suffices or the generalized Fitting subgroup has a...

Although algebraic or so called “implicit polynomial” curves/surfaces have been studied rather extensively for several decades, to the best of our knowledge, a dynamic formulation of them, similar to active contours, has not been done yet. This paper proposes to use implicit polynomial functions instead of signed distance functions in classical level set formulation and shows potential of such ...