Volumes of n-simplices with vertices on a polynomial space curve

In this paper, we give a formula for the area of the triangle formed by the vertices that live on a given polynomial, and we generalize this formula to the volumes of $n$-simplices with vertices on a polynomial space curve. To prove these results, we use induction arguments and a well known identity for complete symmetric polynomials.Comment: 6 pages, 1 figure. The results and the proofs are elementary. Most of the proofs based on induction. However, it is interesting to see how the formulas line up, and can be generalized to higher levels one by on

Generalized Foster's identities

Foster's network theorems and their extensions to higher orders involve resistance values and conductances. We establish identities concerning voltage values and conductances. Our identities are analogous to the extended Foster's identities.Comment: A few minor changes concerning references are made. One of the referenced paper is replaced by a new one. The latest versio

A fast elementary algorithm for computing the determinant of toeplitz matrices

In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix and $k$ is the bandwidth size. This is possible because such a determinant can be expressed as the determinant of certain parts of $n$-th power of a related $k \times k$ companion matrix. In this paper, we give a new elementary proof of this fact, and provide various examples. We give symbolic formulas for the determinants of Toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when $k$ is small.Comment: 12 pages. The article is rewritten completely. There are major changes in the title, abstract and references. The results are generalized to any Toeplitz matrix, but the formulas for Pentadiagonal case are still include